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Full text of "Wave Generation & Shaping"

WAVE GENERATION AND SHAPING 



* 



McGRAW-HILL ELECTRICAL AND ELECTRONIC 
ENGINEERING SERIES 

Frederick Emmons Terman, Consulting Editor 
W. W. Harman, Hubert Heffner, and J. G. Truxal, Associate Consulting Editors 



AHRENDT AND SAVANT Servomechanism Practice 

ANGELO Electronic Circuits 

ASELTINE Transform Method in Linear System Analysis 

ATWATER Introduction to Microwave Theory 

BAILEY AND GAULT Alternating-current Machinery 

BERANEK Acoustics 

BRENNER AND JAVID Analysis of Electric Circuits 

BROWN Analysis of Linear Time-invariant Systems 

BRUNS AND SAUNDERS Analysis of Feedback Control Systems 

CAGE Theory and Application of Industrial Electronics 

CAUER Synthesis of Linear Communication Networks 

CHEN The Analysis of Linear Systems 

CHEN Linear Network Design and Synthesis 

CHIRLIAN AND ZEMANIAN Electronics 

CLEMENT AND JOHNSON Electrical Engineering Science 

COTE AND OAKES Linear Vacuum-tube and Transistor Circuits 

CUCCIA Harmonics, Sidebands, and Transients in Communication Engineering 

CUNNINGHAM Introduction to Nonlinear Analysis 

EASTMAN Fundamentals of Vacuum Tubes 

EVANS Control-system Dynamics 

FEINSTEIN Foundations of Information Theory 

FITZGERALD AND HIGGINBOTHAM Basic Electrical Engineering 

FITZGERALD AND KINGSLEY Electric Machinery 

FRANK Electrical Measurement Analysis 

FRIEDLAND, WING, AND ASH Principles of Linear Networks 

GEPPERT Basic Electron Tubes 

GHOSE Microwave Circuit Theory and Analysis 

GREINER Semiconductor Devices and Applications 

HAMMOND Electrical Engineering 

HANCOCK An Introduction to the Principles of Communication Theory 

HAPPELL AND HESSELBERTH Engineering Electronics 

HARMAN Fundamentals of Electronic Motion 

HARMAN Principles of the Statistical Theory of Communication 

HARMAN AND LYTLE Electrical and Mechanical Networks 

HARRINGTON Introduction to Electromagnetic Engineering 

HARRINGTON Time-harmonic Electromagnetic Fields 

HAYT Engineering Electromagnetics 

HILL Electronics in Engineering 

HUELSMAN Circuits, Matrices, and Linear Vector Spaces 

JAVID AND BRENNER Analysis, Transmission, and Filtering of Signals 

JAVID AND BROWN Field Analysis and Electromagnetics 

JOHNSON Transmission Lines and Networks 

KOENIG AND BLACKWELL Electromechanical System Theory 

KRAUS Antennas 

KRAUS Electromagnetics 



KUH AND PEDERSON Principles of Circuit Synthesis 
LEDLEY Digital Computer and Control Engineering 
LEPAGE Analysis of Alternating-current Circuits 
LEPAGE AND SEELY General Network Analysis 
Jf J,V ¥ L tY T ?; t AND REHBER G Linear Circuit Analysis 
^Im^^ GIBB0NS Transis t°rs ^d Active Circuits 
™™ t™ TRUXAL Introductory System Analysis 
^v^S A ™ ™UXAL Principles of Electronic Instrumentation 

MIU £ AN VD V TRUXAL , SigM,S aDd SyStemS iD E,ectrical Engineering 

^I™ A ^ . y acuura - tub « and Semiconductor Electronics 

MILLMAN AND SEELY Electronics 

MILLMAN AND TAUB Pulse and Digital Circuits 

MISHKIN AND BRAUN Adaptive Control Systems 

MOORE Traveling-wave Engineering 

NANAVATI An Introduction to Semiconductor Electronics 

„tZZII . Electronic Switching, Timing, and Pulse Circuits 

PFEIfJf* 1 McW Q H ORTER Electronic Amplifier Circuits 

ytt.lt thti Linear Systems Analysis 

REZA An Introduction to Information Theory 

REZA AND SEELY Modern Network Analysis 

ROGERS Introduction to Electric Fields 

RYDER Engineering Electronics 

o£^ AR J Z Information Transmission, Modulation, and Noise 

bh.&L,Y Electromechanical Energy Conversion 

SEELY Electron-tube Circuits 

SEELY Electronic Engineering 

SEELY Introduction to Electromagnetic Fields 

SEELY Radio Electronics 

SEIFERT AND STEEG Control Systems Engineering 

SIEGMAN Microwave Solid State Masers 

SISKIND Direct-current Machinery 

SKILLING Electric Transmission Lines 

SKILLING Transient Electric Currents 

SPANGENBERG Fundamentals of Electron Devices 

SPANGENBERG Vacuum Tubes 

STEVENSON Elements of Power System Analysis 

STEWART Fundamentals of Signal Theory 

STORER Passive Network Synthesis 

STRAUSS Wave Generation and Shaping 

TERMAN Electronic and Radio Engineering 

TERMAN AND PETTIT Electronic Measurements 

THALER Elements of Servomechanism Theory 

™ *™ B *°™ Ana,ySiS 8Dd DesigD ° f Feedback Contr °> S * st <>n* 
THALER AND PASTEL Analysis and Design of Nonlinear Feedback Control 
Systems 

THOMPSON Alternating-current and Transient Circuit Analysis 

TOU Digital and Sampled-data Control Systems 

TRUXAL Automatic Feedback Control System Synthesis 

VALDES The Physical Theory of Transistors 

VAN BLADEL Electromagnetic Fields 

WEINBERG Network Analysis and Synthesis 

WILLIAMS AND YOUNG Electrical Engineering Problems 



BROOKLYN POLYTECHNIC INSTITUTE SERIES 



angelo ■ Electronic Circuits 

lynch and truxal • Signals and Systems in Electrical Engineering 
Combining Introductory System Analysis 

Principles of Electronic Instrumentation 

mishkin and braun ■ Adaptive Control Systems 

Schwartz • Information Transmission, Modulation, and Noise 

stbauss ■ Wave Generation and Shaping 



Wave Generation and Shaping 



LEONARD STRAUSS 

ASSOCIATE PROFESSOR OF ELECTRICAL ENGINEERING 
POLYTECHNIC INSTITUTE OF BROOKLYN 



INTERNATIONAL STUDENT EDITION 

McGRAW-HILL BOOK COMPANY, INC. 

New York Toronto London 

KOGAKUSHA COMPANY, LTD. 

Tokyo 



WAVE GENERATION AND SHAPING 



INTERNATIONAL STUDENT EDITION 

Exclusive rights by Kogakusha Co., Ltd. for manufacture and export 
from Japan. This book cannot be re-exported from the country to 
which it is consigned by Kogakusha Co., Ltd. or by McGraw-Hill Book 
Company, Inc. or any of its subsidiaries. 

I 



Copyright © 1960 by the McGraw-Hill Book Company, Inc. All 
rights reserved. This book, or parts thereof, may not be reproduced 
in any form without permission of the publishers. Library of Con- 
gress Catalog Card Number 60-9861 



TOSHO PRINTING CO., LTD., TOKYO. JAPAN 



TO ROSEMARY 



PREFACE 



Time and time again during the last fifty years the electrical engineer 
has been called upon to generate a signal having some specified geometry. 
Either through intuition and experimentation or on the basis of a detailed 
analysis of the specific problem, the first crude circuits were designed, 
improved, and modified. With the advent of the transistor, not only 
were many of the vacuum-tube circuits adapted, but new ones, making 
use of the special transistor characteristics, were devised. Thus today 
hundreds of oscillators, multivibrators, and linear sweeps are described 
in the literature; usually each one is treated as a separate entity. 

When only crude methods of analysis were at the disposal of the 
engineer and when the number of useful active circuits was small, it was 
necessary for the texts to discuss them all in detail. But with the current 
state of the art, an encyclopedic study becomes prohibitive, if, in fact, 
such a work would even be desirable. 

The objective of this text is to present a logical, unified approach to 
the analysis of those circuits where the nonlinearity of the tube or 
transistor is significant. A developmental treatment is followed through- 
out as we focus on the essential features of practical wave-generating 
and -shaping circuits. To this end, the text is arbitrarily divided into 
five sections: models and shaping, timing, switching, memory, and 
oscillations. It is, of course, impossible not to step across these bounds 
in discussing specific circuits and examples have been chosen so that the 
basic ideas will arise naturally from the discussion. In most cases the 
analysis is sufficiently detailed so that the techniques may be applied to 
other existent, and not as yet existent, circuits. Transistors and vacuum 
tubes are used almost interchangeably to support the contention that the 
basic mode of operation is independent of the active element employed. 

The organization, which is described in some detail below, is a result 
of experimentation with class notes over several years. Because this 
work is primarily a text, designed for a senior or graduate course of one 
or two semesters, it is assumed that the reader is familiar with the tran- 
sient analysis of linear networks and with simple vacuum-tube and 
transistor amplifiers. However, since the viewpoint presented may 
be somewhat different from that previously encountered, some review 



X PREFACE 

material is incorporated in the body of the text at the point where it is 
first needed. 

Part 1. Models and Shaping. Since we are interested in the transient 
response of the piecewise-linear models to various excitations, the first 
chapter presents approximation methods which can be employed for 
the rapid solution of linear circuits. 

In Chapter 2 a grossly nonlinear element, the diode, is introduced. 
Once its volt-ampere characteristic is approximated by an ordered series 
of line segments, linear algebraic equations may be written to define the 
behavior in each region. Most of the problems met in more complex 
circuits make their initial appearance when the diode is combined with an 
energy-storage element. Combinations of diodes and resistors can be 
used to represent, in a piecewise manner, almost any arbitrary character- 
istic. For solution, the complex circuit is reduced to a sequence of linear 
networks and the complex problem reduces to finding the boundaries 
between the various regions (break-point analysis). Chapter 3 reinforces 
the concepts of Chapter 2 with additional examples of multiple-diode 
networks. 

The piecewise-linear model is extended to multiterminal devices (the 
tube and transistor) in Chapter 4, relying on the conduction or back 
biasing of ideal diodes to differentiate the regions of operation. In 
order to represent the inherent amplification of the active element, the 
model includes one or more controlled sources. 

Part 2. Timing. The next four chapters deal with the linear sweep. 
Since the essential portion of any timed circuit is the exponential charging 
of an energy-storage element, the problem of analysis resolves itself 
into finding the time-constant, the initial, the steady-state, and the final 
sweep voltage. 

In the simple voltage sweep of Chapter 5, a gas tube is used as the 
switching element. This choice makes it possible to treat synchroniza- 
tion without having recourse to the more complex circuit configurations 
of the later chapters. 

Linearization of the sweep always involves some form of feedback. 
The aim is to approximate constant-current charging of the sweep capaci- 
tor. Vacuum-tube circuits are analyzed in Chapter 6, but a discussion 
of the transistor equivalents is deferred until Chapter 7. In the Miller 
sweep of Chapter 6, the calculation of the initial jumps, overshoots, and 
recovery exponentials, from the individual circuit models, lays the 
groundwork for the later analysis of the multivibrator and the blocking 
oscillator. Furthermore, the separate consideration of the recovery time 
and the switching problem leads to the construction of a sweep system, the 
phantastron. Chapter 8 applies the same techniques of analysis to the 
current sweep. 



PREFACE XI 

Part 3. Switching. When a closed-loop system contains active ele- 
ments and when the loop gain is positive and greater than unity, we 
call it a switching circuit. If any timing networks are included in the 
transmission path, the circuit exhibits quasi-stable behavior. The same 
effect can be obtained by shunting a negative-resistance device with an 
energy-storage element. 

Chapters 9 and 10 discuss a closed-loop regenerative switching circuit, 
the multivibrator. We are mainly concerned with where and how to 
begin the analysis. A guess serves as the convenient starting point, and 
the circuit calculations are checked to see if they yield consistent results. 
If a contradiction arises, it simply indicates that the first guess was wrong. 
In the multivibrator both tubes and transistors are active only during the 
switching interval. At all other times one is cut off or the other is 
saturated. Hence the exponential timing, which sustains the limiting, 
occurs within the cutoff zone, and recovery usually depends on the 
saturated tube or transistor. After the appropriate models are drawn, 
the calculations involve finding the initial and switching points of the 
timing network. 

Many devices exist whose driving-point volt-ampere characteristic 
exhibits a negative-resistance region. The most useful treatment of 
switching circuits containing these elements is from the viewpoint 
expressed in Chapter 11. Switching and timing are first treated with the 
aid of a postulated ideal device. Even though the waveshapes and 
trajectories obtained are approximately correct, the use of an ideal 
negative resistance leads directly to a stability criterion that is incon- 
sistent with the physical device. Since the reasons that the contradictory 
results arose are pointed out later, this chapter also illustrates the dangers 
inherent in overidealizing a model. 

The ideas presented in Chapter 11 may well be considered as unifying 
concepts. These are presented rather late in the text so that the student 
may better appreciate the limitations as well as the advantages of this 
viewpoint. The negative-resistance treatment is useful for understand- 
ing the operation of all sweeps and switching circuits even where it is 
difficult to isolate the two terminals across which the negative resistance 
is developed. It is, however, a convenient design method only for those 
circuits which employ such special devices as the unijunction and p-n-p-n 
transistors and the tunnel diode. 

Not all circuits can, in fact, be solved completely. But the methods 
developed in the earlier sections of the text, when applied to one which 
cannot be, for example, the blocking oscillator of Chapter 12, do yield a 
considerable understanding of how the circuit behaves. 

Part 4. Memory. Since a hysteresis loop indicates memory, magnetic 
and dielectric materials are ideally suited for information storage. 



xii PREFACE 

Chapter 13 examines briefly the terminal characteristics of a core con- 
structed out of ideal square-loop material. Furthermore, a clearly 
defined time is needed to saturate the core, and although not representable 
in a piecewise-linear manner, the driving-point impedance exhibits two 
distinct regions. This enables us to use either a core or a ferroelectric 
device as the timing element in a switching circuit. 

Part 5. Oscillations. Although most of the circuits previously dis- 
cussed depended for their timing on a single energy-storage element, at 
least two are necessary in order for the system to have a pair of complex 
conjugate poles located in the right half plane. 

The sinusoidal oscillator is first treated as an almost linear feedback 
system (Chapter 14) which allows the separation of the frequency-, 
amplitude-, and gain-determining element. The role of each oscillator 
essential is individually examined, with emphasis on minimizing the 
distortion and maximizing the stability. Since the signal produced is 
almost sinusoidal, the amplitude is readily found by plotting a system- 
describing function. 

Chapter 15 returns to the negative-resistance viewpoint and describes 
the solution of a specific nonlinear differential equation. Here topological 
constructions, which supplement analytic methods, yield the waveshape 
and amplitude of oscillations. Hence the text ends by introducing a new 
topic and not by saying the last word on an old one. 

While preparing the manuscript, I received a great deal of assistance 
and encouragement from many people. I should like to acknowledge the 
contributions made by those colleagues who taught sections of the course 
for which this text was written. In stimulating technical discussions, I 
received the benefit of their ideas and thinking on many topics, much of 
which caused me to modify the original treatment. They also corrected 
errors which inadvertently crept into the original class notes, and they 
prepared some of the problems. The secretarial and drafting assistance 
made available by the Polytechnic Institute of Brooklyn is greatly 
appreciated. Finally, I wish to thank all members of the Electrical 
Engineering Department of the Institute for their generous encourage- 
ment during the project. The friendly and cooperative atmosphere made 
the writing of this book almost a pleasure. 

Leonard Strauss 



CONTENTS 

Preface ix 

PART 1— MODELS AND SHAPING 

Chapter 1. Linear Wave Shaping 3 

1-1. Introduction 3 

1-2. Initial Conditions 3 

1-3. Solution of Periodically Excited Circuits 8 

1-4. Differentiation 10 

1-5. Integration 12 

1-6. Summation 13 

1-7. Approximate Solutions of the Singkr-energy Case 14 

1-8. Double-energy-storage-element Systems 16 

1-9. Compensated Attenuators 19 

Chapter 2. Diode Wave-shaping Techniques 26 

2-1. Passive-nonlinear-circuit Representation 26 

2-2. Physical Diode Model 32 

2-3. Voltage Clipping Circuits 33 

2-4. Single Diode and Associated Energy-storage Element — Some General 

Remarks 37 

2-5. Diode Voltage Clamps 38 

2-0. Current Clippers 45 

2-7. Current Clamps 46 

2-8. Arbitrary Transfer and Volt-Ampere Characteristics 48 

2-9. Dead Zone and Hysteresis 55 

2-10. Summary 59 

Chapter 3. Diode Gates 67 

3-1. Application 67 

3-2. or Gate 67 

3-3. and Gate 72 

3-4. Controlled Gates 75 

3-5. Diode Arrays 78 

3-6. Diode Bridge — Steady-state Response 79 

3-7. Diode Bridge — Transient Response 85 

3-8. Concluding Remarks on Diode Gates 88 

Chapter i. Simple Triode, Transistor, and Pentode Models and Circuits . 96 

4-1. Triode Models 96 

4-2. Triode Clipping and Clamping Circuits 107 

4-3. Triode Gates . 110 

4-4. Transistor Models 116 

4-5. Simple Transistor Circuits 125 

xiii 



XIV CONTENTS 

4-6. Transistor Gates 130 

4-7. Pentodes 133 

4-8. Summary 135 

PART 2— TIMING 

Chapter 6. Simple Voltage Sweeps, Linearity, and Synchronization 147 

6-1. Basic Voltage Sweep 147 

5-2. Gas-tube Sweep 148 

5-3. Thyratron Sweep Circuits 152 

5-4. Sweep Linearity 154 

5-5. Synchronization 157 

5-6. Regions of Synchronization 161 

Chapter 6. Vacuum-tube Voltage Sweeps 172 

6-1. Introduction 172 

6-2. Linearity Improvement through Current Feedback 172 

6-3. Bootstrap Sweep 175 

6-4. Miller Sweep 178 

6-5. Recovery-time Improvement . 186 

6-6. Sweep Switching Problems 189 

6-7. Pentode Miller Sweep 190 

6-8. The Phantastron 194 

6-9. Miller Sweep and Phantastron — Screen and Control-grid Voltage Calcu- 
lations 198 

Chapter 7. Linear Transistor Voltage Sweeps 206 

7-1. Constant-charging-current Voltage Sweep 206 

7-2. Bootstrap Voltage Sweep 209 

7-3. Miller Sweep 214 

7-4. Compound Transistors 216 

Chapter 8. Linear Current Sweeps 222 

8-1. Basic Current Sweeps 222 

8-2. Switched Current Sweeps 226 

8-3. Current-sweep Linearization 232 

8-4. A Transistor Bootstrap Sweep 235 

8-5. Constant-current-source Current Sweep 238 

PART 3— SWITCHING 

Chapter 9. Plate-Grid- and Collector-Base-coupled Multivibrators 247 

9-1. Basic Multivibrator Considerations 247 

9-2. Vacuum-tube Bistable Multivibrator 249 

9-3. Transistor Bistable Multivibrator 255 

9-4. A Monostable Transistor Multivibrator 258 

9-5. A Vacuum-tube Monostable Multivibrator 261 

9-6. Vacuum-tube Astable Multivibrator 260 

9-7. Transistor Astable Multivibrator , . . . . 269 

9-8. Inductively Timed Multivibrators 274 

9-9. Multivibrator Transition Time 277 

9-10. Multivibrator Triggering and Synchronization 280 



CONTENTS XV 

Chapter 10. Emitter -coupled and Cathode-coupled Multivibrators .... 289 

10-1. Transistor Emitter-coupled Multivibrator — Monostable Operation . . 289 

10-2. Modes of Operation of the Emitter-coupled Multivibrator .... 293 

10-3. Monostable Pulse Variation 301 

10-4. Emitter-coupled Astable Multivibrator 303 

10-5. Cathode-coupled Monostable Multivibrator 307 

10-6. Limitation of Analysis 310 

Chapter 11. Negative-resistance Switching Circuits 315 

11-1. Basic Circuit Considerations 315 

11-2. Basic Switching Circuits 317 

11-3. Calculation of Waveshapes 321 

11-4. Voltage-controlled NLR Switching Circuits 325 

11-5. NLR Characteristics — Collector-to-base-coupled Monostable Multi- 
vibrator 326 

11-6. Some Devices Possessing Current-controlled NLR Characteristics 329 

11-7. Some Devices Possessing Voltage-controlled Nonlinear Characteristics. 336 

11-8. Frequency Dependence of the Devices Exhibiting NLR Characteristics 341 

11-9. Improvement in Switching Time through the Use of a Nonlinear Load . 346 

11-10. Negative-impedance Converters 349 

Chapter 12. The Blocking Oscillator 357 

12-1. Some Introductory Remarks 357 

12-2. An Inductively Timed Blocking Oscillator 359 

12-3. Transformer Core Properties and Saturation ....:... 365 

12-4. Capacitively Timed Blocking Oscillator 367 

12-5. Astable Operation 370 

12-6. A Vacuum-tube Blocking Oscillator 372 

12-7. Some Concluding Remarks 380 

PART 4— MEMORY 

Chapter 13. Magnetic and Dielectric Devices as Memory and Switching Ele- 
ments 387 

13-1. Hysteresis — Characteristics of Memory 387 

13-2. Ferromagnetic Properties 389 

13-3. Terminal Response of Cores 393 

13-4. Magnetic Counters 400 

13-5. Core-transistor Counters and Registers 406 

13-6. Magnetic Memory Arrays 408 

13-7. Core-transistor Multivibrator 410 

13-8. Properties of Ferroelectric Materials 414 

13-9. Ferroelectric Terminal Characteristics 417 

13-10. The Ferroelectric Counter 419 

13-11. Coincident Memory Arrays 421 

PART 5— OSCILLATIONS 

Chapter 14. Almost Sinusoidal Oscillations — The Linear Approximation . 431 

14-1. Basic Feedback Oscillators 431 

14-2. Characteristics of Some RC and LR Frequency-determining Networks . 435 

14-3. Transistor Feedback Oscillators 442 



XVi CONTENTS 

14-4. Tuned-circuit Oscillators 445 

14-5. Frequency Stabilization 455 

14-6. Amplitude of Oscillations 460 

14-7. Amplitude Stability 471 

Chapter 15. Negative-resistance Oscillators 477 

15-1. Basic Circuit Considerations 477 

15-2. First-order Solution for Frequency and Amplitude 482 

15-3. Frequency of Oscillation to a Second Approximation 487 

15-4. Introduction to Topological Methods 492 

15-5. The Lienard Diagram 499 

15-6. Summary 505 

Name Index 511 

Subject Index 513 



PART 1 

MODELS AND SHAPING 



CHAPTER 1 

LINEAR WAVE SHAPING 



1-1. Introduction. Often the control and instrumentation engineer 
finds himself faced with the problem of producing a designated complex 
waveform. Usually his simplest approach is to divide the problem into 
two parts: first, the generation of simple waveshapes and, second, some 
type of operation on them to achieve the desired final result. If the 
required waveform is of a sufficiently complex nature, several substages 
of shaping may be needed, with combination taking place in the final 
stage. Generation always involves the use of active elements, and 
although they also perform a function, shaping is predominantly effected 
by the circuit's passive components. One might even go so far as to say 
that a major portion of the process of wave generation itself consists in 
passive element shaping. 

In this chapter we shall examine the behavior of linear passive circuits 
with a view to solving for their time response to various input wave- 
shapes. Exact analysis is, of course, possible, but often so tedious that 
the competent engineer looks instead for reasonable approximations. 
Our aim, therefore, is to set up conditions whereby solutions may be 
obtained rapidly, almost by inspection, avoiding involved algebraic 
manipulation. This, in fact, becomes possible only under very specific 
circumstances, which must be clearly delineated. Particular emphasis 
will be placed on the treatment of circuits containing only a single-energy- 
storage element, not only because these appear quite frequently, but also 
because it is here that the approximation techniques are most fruitful. 
Sometimes even multiple-energy-storage circuits may be reduced to a 
number of separate single-energy circuits, which are then individually 
solved. The solutions are finally combined, yielding the approximate 
total response of the original complex circuit. 

In our effort to develop facility in utilizing simplifying approximations, 
i.e., how to make them and when to make them, we shall start by con- 
sidering some very basic circuits. Since we already know methods of 
finding their exact response to simple inputs, we can concentrate atten- 
tion on the methodology of the solution. 

1-2. Initial Conditions. We shall first consider the transient response 
of a passive linear network, i.e., the response to some abrupt change in 

3 



4 MODELS AND SHAPING [CHAP. 1 

either the external excitation or the internal circuit. We know that 
the solution will be directly related to the poles of the system transfer 
function, and for poles at pi, p 2 , and p 3) the response to a step will always 
have the form 

r(t) = A + Aie>»' + A i e'" t + A z e">' 

One exponential term is due to each energy-storage element included 
in the network. This type of response is readily found by the classical 
method of writing and solving the system's integrodifferential equations 
or by modern transformation analysis. 

If, however, we restrict our interest to relatively simple systems, i.e., 
those containing one, or sometimes two, widely separated poles, then 
these may often be found by direct inspection of the circuit. They will 
correspond to the negative reciprocal of the time constants. 



20v:=: 



Fig. 1-1. Single-energy-storage-element circuit. 

In a circuit containing only a single-energy-storage element, such as 
the one shown in Fig. 1-1, the response to a unit step will be given by 




e (t) = A + Be-"' 



(1-D 



The first parameter of interest, the time constant, is simply the product 
of C and the total resistance seen by the capacitor. When the switch 
is in position 2, 

t 2 = (fli + R.)C p 2 = - r (1-2) 



T 2 

Upon switching to position 3, both the time constant and pole location 
change; they now become 

r 3 = RxC V z = - - (1-3) 

The time response is of the form given in Eq. (1-1) ; coefficients, of course, 
will differ. 

Since a prerequisite to the complete solution is the value of the various 
coefficients, a slight digression as to the means of their evaluation is in 
order before proceeding to the general solution of this circuit. But 
these coefficients depend upon the initial conditions; therefore the first 



Sec. 1-2] lineab wave shaping 5 

question to ask is, What determines these conditions? Obviously, one 
part of the answer is the known external constraints imposed on the 
circuit, i.e., the excitation function and the various circuit changes. The 
second half of the answer is the internal constraints determined by the 
energy-storage elements present within the circuit, in this case a single 
capacitor. 

The voltage drop across any capacitor is related to circuit current flow 
as expressed in the differential equation 

e c =±fidt = Q (1-4) 

This equation says that, in a physical system, the accumulation of charge 
takes finite time unless an infinite current flows. Consequently the 
terminal voltage cannot change instantaneously. In an ideal system 
such instantaneous changes may be forced by placing a short circuit or 
an ideal battery across the capacitor terminals or by injecting an infinite 
pulse of current. These do not exist in practice, but sometimes the 
charge or discharge time is so very fast compared with the total time 
range of interest that the small time interval involved may be ignored, 
and we can say that the capacitor voltage has jumped to its final value. 
From the above discussion, the first circuit constraint can now be stated : 
the voltage across the capacitor must be continuous across regions of circuit 
or excitation change. Carrying this argument a step further leads us 
to the additional conclusion that all sudden changes in circuit voltage 
will be distributed across the various noncapacitive portions of the 
circuit — across the resistors and inductances. The manner of division 
depends on the particular circuit configuration under investigation. 

One can derive a second condition from Eq. (1-4) by noting that the 
voltage across C continues to change as long as any current flows. The 
final steady-state voltage is reached when i c = 0. In a simple network 
it will always be the d-c Thevenin equivalent voltage appearing across 
the capacitor. 

We conclude that the constant A in Eq. (1-1) represents the steady- 
state response and may be evaluated by considering the behavior at 
t = » . The coefficient B is related to the behavior at t = and would 
be found from the initial conditions. 

As an example, consider the time response of the circuit of Fig. 1-1. 
Its behavior is the following: at t = the switch is thrown from position 1 
to 2; once the voltage across the capacitor reaches the predetermined 
final value, Ef = 15 volts, the switch is moved to position 3, remaining 
there. 

Rather than attempt to solve the entire problem in a single step, it is 
advisable to divide it into three simple parts and to solve each in proper 



6 



MODELS AND SHAPING 



[Chap. I 



time sequence. The complete solution will then be the aggregate of the 
individual answers. The three parts are: 

1. Behavior at t = - 

2. Response from I = 0+ to the time t t at which e„(t/) = E t 

3. Response from t = t/ to » 

Note that the change of state determines the problem division, with 
each part holding for a specific circuit configuration. In proceeding, the 
solution of each part will start from its own zero time as if it were a com- 
pletely independent problem. Numerical subscripts corresponding to 
the particular part indicate the time region under examination. 

1. Solution for t < Oi: 

e i = — 10 volts 

2. System response for (h < t < t/2. In this region the solution is 
given by Eq. (1-1). We are now ready to evaluate the coefficients from 
the specific initial conditions holding. 





20 












15 






** 
* 














t 








1 \r 3 




*o(0 






V 











/ 




*/2 
1.79r 2 


t 


E a 


-10 











Fig. 1-2. Total time response of the circuit of Fig. 1-1. 

Because the voltage of C must be continuous across the switching 
interval (circuit change), the initial value in this region (En) will be the 
same as the final value of part 1 (E/i). At t = + , 



e„ 2 (0+) = E it = E f i = -10 volts = A 2 + B 2 



(1-5) 



Furthermore, the output now charges toward a steady-state value E,,i, 
found at t = » 

e<, 2 (°°) = E„i = 20 volts = A 2 (1-6) 

Solving Eqs. (1-5) and (1-6) and substituting into (1-1) yields 

e 02 (<) = E„2 - (E.. t - E i2 )er<i" 

= 20 - 30e-« r > (1-7) 

The final value of part 2 is found by substituting t fi into Eq. (1-7). 



Sec. 1-2] linear wave shaping 7 

e i(t/ t ) = E f i = 15 volts = E..2 - (E„ t - E it )e-'»i r * 

Solving for t/i yields 

, , E„2 — En , 20 — ( — 10) t _ n ,, . 

tfi = Ti ln E„ t - E n = T2 ln 20-15 = L79t2 (1 ' 8) 

3. System response for < /2 < t < °o or 3 < t < °o. By the same 
methods used for part 2, the new defining equation becomes 

(1-9) 







e„ 3 (0 = E.,3 


- (E.. % - E i3 )e-"" 


The 


t initial and 


steady-state voltages in 


region 3 are 


(a) 




t = 3 


En = 


Efi = 15 


(b) 




t = 00 


e s = 


E.,3 = 



Substituting these conditions into Eq. (1-9) results in the final decay 
equation 

e oS (0 = E/tfT'i" = 15e-"" 

Figure 1-2 is the plot of the total solution, with the response of each 
portion beginning immediately after that of the previous part. 

In the solution for part 2, the circuit responded as if no change were 
scheduled. It could not predict the future, and therefore, in solving 
the problem, we must be careful lest our foreknowledge lead to fallacious 
reasoning, i.e., the incorrect substitution of E/t instead of E„ 2 for the 
steady-state value. Upon entering region 2 the response is toward the 
final value calculated as if the circuit were invariant. 

When the circuit contains an ideal inductance, additional constraints 
appear. 

•L-ig (MO) 

From Eq. (1-10) it may be seen that an infinite voltage is required to 
change the current instantaneously. As this is an impossible situation 
in any physical circuit, our first conclusion is that the current through 
the coil must be continuous across circuit or excitation changes. A second 
constraint, also derivable from the above equation, is that steady state 
is finally reached when there is no further time rate of change of current, 
i.e., when ex, = 0. These conditions are just the duals of the ones found 
for the capacitor. The overlapping constraints of the two energy- 
storage elements forbid any instantaneous changes of current, and of 
capacitor or resistor voltage, in a series RLC circuit. Because of the 
physical nature of the various circuit elements, they will all have associ- 
ated stray capacity and lead inductance. Therefore all circuits contain 



8 



MODELS AND SHAPING 



[Chap. 1 



more than one type of energy-storage element. These parasitic elements 
act to slow down the rapid changes found when we consider a simple 
ideal BL or BC circuit. But if the time range of interest is sufficiently 
long, the second-order effects may be neglected. 

1-3. Solution of Periodically Excited Circuits. After the sudden 
application of a periodic input to a passive network, the output itself 
will eventually become periodic in nature. The transient dies out in 
four time constants, leaving the steady-state response. The time 
required for this, in terms of the number of cycles of the input wave- 
shape, depends upon the relative values of the time constant and the 
period of the excitation function. If the time constant is much smaller 
than the input period, the periodic output appears within one or at 
most a few cycles and it can be found by following the behavior cycle by 
cycle. 

R 

9 WV- 



u 

ei(t) 


*6 


Region I 


Region U 




/ 




*-r,— 


T z 


t 



\ 

erf) 



e 2 (t) 



Fig. 1-3. Circuit for the study of recurrent boundary conditions. 



On the other hand, when the ratio of the time constant to the period 
of the input is relatively large, periodicity will not be reached until an 
extremely large number of cycles have passed. During the build-up, 
the capacitor charges to the average level of the voltage appearing across 
its terminals and the direct current through a coil increases from zero 
to the average circuit value. To attempt to start at zero time and follow 
the response cycle by cycle until final periodicity is a ridiculous approach 
to the solution of this class of problems — it will only lead to frustration. 

A more judicious treatment would be the application of the results 
of the previous section, with the addition of the known periodicity of 
the solution. This enables us to write recurrent initial conditions; i.e., 
the starting point of any one cycle is the same as that of the next cycle. 
As an example, we shall now examine the RL circuit of Fig. 1-3. The 
rectangular input shown has been applied for a sufficiently long time 
so that the output has already reached the final periodic form shown in 
Fig. 1-4. 

The solution during either portion of the input signal is 



e t (t) = A + Ber'ir 



L 
R 



Sec. 1-3] 



LINEAR WAVE SHAPING 



9 



Except for the time constant, this equation is of the same form as the 
one found for the RC circuit [Eq. (1-1)]. Since the inductance prevents 
abrupt changes in the current, the resistor voltage will be invariant 
across the discontinuities of the input and any abrupt jump in the driving 
voltage will be reflected across the coil. We can show this by considering 
the voltage drops around the loop: 

ci(0 = e B + e L = iR + e£ (1-11) 

Across the boundary of the two regions the input voltage jumps, and 
by considering this change as Aei(i), we see that 

Ae x (t) = MR + Ae L 

But since the inductance constrains At to be zero, the complete jump 
in the input voltage must immediately appear across the inductance. 




Fig. 1-4. Output voltage across L in the circuit of Fig. 1-3. 

Starting the solution at the beginning of region I, which holds for 
Oi < t < T\, the first set of initial conditions is 

E.,i = En = unknown 

Therefore the solution in this region is 

««(0 = Ear*" (1-12) 

and the final value, occurring at t = T\, may be expressed as 

E n = «■ (TO = Bar*" (1-13) 

Upon entering region II, the input voltage drops from E a to E b , with 
the same change immediately appearing across the inductance. The final 
output of region I was E/i. The initial value of region II must be 

Em = E/i — (E a — Ei,) 



10 MODELS AND SHAPING [CHAP. 1 

In region II, holding for 2 < t < Ti, E„ n = 0. The equation for the 
time response can be written 

«m(0 = [E n ~ (E. - E h )]e-'» (1-14) 

At t = T h 

E m = e ai (T 2 ) = [E n - (E a - E b )]e-™ (1-15) 

But the initial voltage of region I, and all other voltages which have 
been written in terms of it, are still unknown. We must now apply the 
known conditions of periodicity in order to find the final answer. At the 
end of region II the input voltage jumps from E\, to E a and the output 
voltage changes by the same amount. Periodicity tells us that the new 
value upon reentering region I is the original assumed E a . 

E a = E m + (E. - E h ) (1-16) 

Substituting Eq. (1-16) into (1-13) results in 

E n = [E m + {E a - E b )]e-^ir (1-17) 

Equations (1-15) and (1-17) each involve only the two unknowns En and 
Efii', all other terms are constants of the input waveshape or the circuit. 
The simultaneous solution of the above equations completes the analysis. 
If the output of interest were taken across the resistor instead of the 
inductance, the technique of solution would still be the same, the prin- 
cipal difference being in the initial conditions over each portion of the 
cycle. Current is continuous, and therefore resistor voltage must also 
be continuous in the RL circuit. RC series circuits are treated in a 

similar manner, applying, however, 
? l( | ? their special boundary conditions; i.e., 

I < I all voltage jumps appear across the 

* l l R S * 2 | circuit resistance. 

1 | i Even when the inputs are nonrec- 

Fig. 1-5. Differentiator. tangular, the shape and equation of 

their response over any period or 
portion thereof usually can be obtained by many convenient methods. 
The known periodicity supplies the additional information necessary for 
a complete solution. 

1-4. Differentiation. A series RC circuit (Fig. 1-5) will, contingent 
on the satisfaction of certain conditions, have an output approximately 
proportional to the derivative of the input. These conditions may best 
be determined by examination of the circuit differential equation 

e,(t) = Ri = ei(t) - ^ I i dt (1-18) 



Sec. 1-4] linear wave shaping 11 

Since the voltage across a resistor is directly proportional to the current 
flow through it, for the output to be proportional to the derivative, the 
current must also have this relationship. Satisfaction of this necessary 
condition is made possible only by maintaining the output voltage small 
with respect to the input. Then from Eq. (1-18), 



Therefore 
and 



ei(0 £g j idt 
, dei(t) 



i^C- 



csS-RC 



dt 

deijt) 
dt 



(1-19) 



The major portion of the circuit voltage drop is developed across C 
only when the time constant is small compared with the time range of 
interest. At discontinuities the total change of input voltage appears 
at the output and the circuit only roughly approximates a differentiator. 




19.6 
•2ft) 

-J 

-19.6 



i 



V T-200^«sec 



r 



Fig. 1-6. Differentiator input and output waveshapes. 



With a sufficiently small time constant, the exact output waveshape is 
reasonably close, except at the discontinuities, to the one found by assum- 
ing a perfect differentiator. The exact derivative at a discontinuity 
of the input waveshape is infinity. Since instantaneous jumps do not 
exist in nature, the full change in the input voltage, which appears across 
the output at the input "discontinuities," is so very much larger than the 
normal small output signal that the circuit may well be said to approxi- 
mate the derivative even here. Thus, after checking the time constant 
against the input waveshape, the output can be drawn by inspection and 
the voltage values calculated later. Figure 1-6 illustrates this process 
for an RC circuit having a time constant of 200 /usee. 



12 MODELS AND SHAPING [CHAP. 1 

Because the capacitor will not pass direct current, the periodic response 
to periodic inputs must have zero average value. The capacitor will 
store a charge proportional to the average level of the input signal, and 
its voltage will fluctuate almost between the input extremes. 

An RL series circuit having a small time constant also differentiates. 
Here the output must be taken across the inductance because the basis 
of differentiation in this circuit is the proportionality of the inductive 
voltage to the time rate of the circuit current change. By maintaining 
H almost the full input voltage drop across 

the series resistor, we ensure the cur 



t 



* rent's dependency only on this resist- 
ed C^ e 2 (t) ance and guarantee good differentiation. 

I 1-5. Integration. Provided that the 

° time constant is very long, the capacitor 
Fig. 1-7. Integrator. voltage in a series RC circuit is pro- 

portional to the integral of the input 
voltage (Fig. 1-7). Again the required condition may be found from 
an examination of the circuit differential equation 



-hi 



eiif) = g idt = e x (0 - 'Ri (1-20) 

When the output is small compared with the input, the approximate 
relationship 

ei(<) ^ Ri 
results, and therefore 

e,(.t)^4n ( e > dt (1-21) 



If the output voltage rises above a relatively small value, the current, 
* = (ei — e 2 )/R, becomes dependent on e% and the circuit will no longer 
integrate satisfactorily. An excessively rapid rise in the output voltage 
within any period is directly dependent on the rate of charging of C. 
A slow charging rate remains possible only through keeping the time 
constant long compared with the time over which the integral is desired. C 
eventually charges to the d-c level of the input with the circuit time con- 
stant, and the output voltage shifts accordingly. For a signal impressed 
at t = 0, Fig. 1-8 shows both the input and its integrated output. Even- 
tually the output will shift down until it has a zero average value. 

The output voltage, at any time, is proportional to the area under the 
curve of the input signal from zero to that point. Consequently its 
shape may be sketched directly from the curve and the coordinates found 
from Eq. (1-21). In order to calculate the voltage change at the output 



Sec. 1-6] 



LINEAH WAVE SHAPING 



13 



over any time interval, the input volt-time area is simply divided by the 
time constant. 

A large-time-constant RL circuit also integrates with the small output 
voltage now appearing across R. Physical inductances always have 
associated winding resistance and, for large coils, appreciable stray 





R 2 
VWW 9 



Fig. 1-8. Input signal and integrated output. Input applied at J = 0. 

capacity. The impossibility of obtaining an ideal element usually pre- 
cludes the use of RL circuits for many wave-shaping functions. On the 
other hand, almost ideal capacitors are readily available. Using the new 
plastic dielectrics, leakage resistance, the major parasitic element, is of 
the order of 10 5 megohms, a value 
so large that it may be ignored in 
most circuit calculations. 

1-6. Summation. After independ- 
ent wave-shaping operations have 
been performed on various signals, 
combination is accomplished by sum- 
ming the individual results. Of the 
several available methods, the simplest, shown for two inputs, uses the 
resistor network of Fig. 1-9. From the circuit we see that the two 
input currents are 

e\ — e„ . ej — e 



Fig. 1-9. Summation circuit. 



li = 



Ri 



Rz 



Provided that the output level is kept low compared with each of the 
inputs, the current through each input resistor is approximately inde- 
pendent of the output voltage. But this is possible only if R <K Ri 
and R <K R2. Almost the complete voltage drop appears across the 
input resistors, and 

(1-22) 



»iS 



Ri 



The output becomes 



ti= R t 



e, = R(ii + it) = 



R , R 

W ei + p" e * 
Hi tit 



(1-23) 



«2° ^Sl&Lr 

L 



14 MODELS AND SHAPING [CHAP. 1 

Even when R is not small compared with the other resistors, the out- 
put voltage can still be found by the linear superposition of the indi- 
vidual terms. The circuit operates by converting the input voltage 
into a proportional current and then summing all the individual inde- 
pendent currents. Equation (1-23) shows the output's proportionality 
to the sum of the individual inputs, each of which is multiplied by a scale 
factor. Determination of this factor is through a choice of the input 

resistors Ri and R^. Summation 

£ may be extended to multiple inputs 

1 ' by simply adding additional input 

9 branches. 

. . A ~ J„ The condition for summation, 

e 3 o WW ' < p | 

r 3 £ i.e., the output voltage small com- 

pared with each input, is exactly 

Fig. 1-10. Combined operations (differ- the same as that squired for inte- 
entiation, integration, summation). gration and differentiation, and all 

three operations may conveniently 
be combined in a single circuit as illustrated in Fig. 1-10. The output, 
expressed by Eq. (1-24), is found by calculating and sketching each term 
individually and summing the results; this is often carried out graphically. 

*»-**£ +271 /** + #,•■ (1_24) 

CR must be small and L/R very large compared with the time interval 
of the various signals. The inputs, if desired, may be supplied from the 
same source. 

1-7. Approximate Solutions of the Single-energy Case. The results 
of the previous sections are quite valuable in that they open up two classes 
of single-energy-storage-element circuits to rapid solution. The first 
group consists of those with short time constants, the differentiators; 
the second, those with long time constants, the integrators. These two 
cases, representing only about 10 per cent of the possible range of RC or 
RL circuits, include, however, a much larger portion of the important 
engineering problems. 

In handling a problem of this nature, our first step is to compare 
the time constant with the input signal and on the basis of this comparison 
to classify the circuit response. If it is a differentiator, the voltage across 
the resistor or inductance may be sketched by inspection, and the drop 
across the other circuit element is simply the difference between the 
applied input and the now known branch voltage. In an RC integrator, 
the voltage across C is the easily found quantity, with the resistor voltage 
remaining the single unknown. This is illustrated in Fig. 1-11, where 
the signal is assumed to be applied at zero time and where we shall 



Sec. 1-7] 



LINEAR WAVE SHAPING 



15 



apply the techniques of integration in finding the voltage across both B 
and C. Because C cannot pass direct current and e« has a d-c level, the 



«i 
1 



Hf- 



RC-l msec 



10 

t 

e,(t) 






















-10 





20 


40 


60 


80 


(,/isec 









10 
f 9.8 

eo(0 
0? 


?=—<■— 










-in 











t 


-102 


1 










Fio. 1-11. Integrator circuit and approximate output. 

output waveshapes sketched (Fig. 1-11) require a minor correction. The 
voltages must be shifted slightly until both have zero average value. 
Figure 1-12 shows the final periodic solution. 




-9.9 

-10.1 
Fig. 1-12. Final periodic solution of the problem of Fig. 1-11. 

It is often convenient to be able to express the response of a long-time- 
constant circuit to square or rectangular inputs as an equation. The 
exact solution of any single-energy-storage-element circuit is, of course, 



e„(t) = A + Be-'* 



(1-25) 



Substitution of the series expansion [Eq. (1-26)] for the exponential simpli- 
fies the final result. Since the time constant is long, provided that we 
restrict the ratio of the maximum interval to the time constant, <„/t, 



16 



MODELS AND SHAPING 



[Chap. 1 



to less than 0.1, an error of less than 5 per cent is introduced by the 
neglect of the higher-order terms of the expansion. 

(1-26) 



t 



Therefore, for t/r < 0.1, Eq. (1-25) may be approximated by 

e.(0 Si+ifl-j) 



(1-27) 



which is recognizable as the equation of a straight line having a slope 
-B/t. 

If the exact expression of the response [Eq. (1-25)] were differentiated 
and evaluated at t = 0, the slope found would be the same as that 
obtained from the first term in the series expansion. The interpretation 
which follows is that if the circuit response is restricted in time to the 
beginning of the exponential decay, this small portion of the curve may 
be represented by a straight-line segment. 

1-8. Double-energy-storage-element Systems. Many circuits con- 
taining two energy-storage elements have poles separated widely enough 
so that the transient response can be approximated by treating them as 
two isolated single-energy circuits. Of course there must be continuity 
across the boundary between the two individual response curves and 
they must satisfy the original system. The pole which is located far 
from the origin (small time constant) will determine the circuit's behavior 
with respect to any fast changes in the input excitation. It predom- 
inantly controls the initial rise. The pole which is located close to the 
origin is related to the low-frequency response of the system and will 

contribute a slow exponential to the 
output. It will determine the hold- 
ing power, i.e., the decay rate. 

In effecting the separation of 
the system response into two time 
regions, we shall have to depend on 
the physical characteristics of the 
energy-storage elements for clues as 
to the permissible approximations. 
This technique is best illustrated by 
an example, and to this end we shall analyze the simple BC-coupled 
amplifier whose incremental model is shown in Fig. 1-13. 

Consider the application of a voltage step of height — E at the grid 
of the tube. The output is constrained in its time rate of rise primarily 
by C,. We see from the circuit (Fig. 1-13) that the full charging current 
of C. together with any current through R a must also flow through C e . 




Fio. 1-13. Incremental model for an 
BC-coupled amplifier. 



Sec. 1-8] linear wave shaping 17 

If, as is normally the case, R„ is very large, the total charge flow just 

after the excitation is applied is essentially determined by the uncharged 

shunt capacity C,. The same charge is also accumulated in C e . Since 

C c ~S> C„ the voltage across C c will r p 

change but slightly while C, charges I * v ^ v 

fully. Thus the coupling capacitor JL 

may be assumed to be a short circuit ^ pe^-jiE ^ r l -R*^ C* 5 ^ 4>i 

during this entire interval and the 

equivalent circuit is reduced to one 

containing a single-energy-storage 

element (Fig. 1-14). Fla Uli ' Rise e l uivalent circuit - 

The final steady-state output and the circuit time constant are found 
by taking the The>enin equivalent across C. They are 

Em1 = r p + L R L f\'R t E ri = C ' {R ° H Rl l! r *> (1 ' 28) 

The parallel lines indicate that the adjacent elements are in parallel; 
for example, R L || R g = R L R g /(R L + R g ). This notation will be used 
throughout in an attempt to keep the circuit-element relationships 
apparent in any equations which will be written. 

We are now in a position to write the equation defining the initial 
portion of the output response. 

e„i = -B»,i(l - e-«"0 (1-29) 

In four time constants, the output rises to 98 per cent of the steady-state 
value E„i and the initial rise may be assumed complete. 

During this whole interval, C c is charging, even though it is doing 
so very slowly. The relatively large current required (because of the 
large value of C c ) will now control the output voltage and swamp any 
contribution from the discharge of C,. We are justified in ignoring C, 
and in removing it from the circuit. If the initial value of the output 
across R g , upon the sudden excitation of the system, is now calculated, 
we also find it to be E Ml . As C c charges, the output decays toward zero, 
with the new time constant 

t 2 = C c {R g + R L || r P ) (1-30) 

Thus the equation defining the final portion of the response may be 
written by inspection. 

«o2 = E, tl e- l ' r ' (1-31) 

The only question remaining unanswered is, At what point will the 
decay take over from the initial rise? If we compare Eqs. (1-28) and 
(1-30), we see that both the capacity and the resistance terms of n 



18 



MODELS AND SHAPING 




Fig. 1-15. flC-coupled amplifier out- 
put — step excitation. 



[Chap. 1 

are much larger than n. Since the decay time constant is so very much 
longer, the error introduced by starting the decay anywhere in the 
vicinity of zero will be negligible. We might just as well choose the 
most convenient point, the end of the initial rise, that is, 4n. The total 
output is sketched in Fig. 1-15. 

The exact solution, the dashed line, shows the greatest deviation at the 
peak asymptotically approaching the approximate solution at both long 

and short times. By the very nature 
of the assumption made, i.e., com- 
plete charge before any discharge, 
the exact answer will always lie 
below the approximate one. As 
the separation of time constants in- 
creases, agreement between the two 
answers improves. For r 2 > IOOti, 
as is normal in this type of amplifier, 
agreement is extremely good. 
When an amplifier is used in pulse applications, the time response, 
rather than the related high-frequency 3-db point, is used to characterize 
its quality. Even if the amplifier transmits all signals faithfully, it may 
still delay them by a fixed time interval ; this delay will also be considered 
as a defining quantity. Generally, the shape of the output will be 
affected, and in order to separate the rise time from the inherent delay, 
we choose the 10 per cent value as our lower reference point. Moreover, 
the applied pulse makes its presence known long before the output reaches 
steady state; the 90 per cent point is therefore arbitrarily chosen as the 
upper reference value. An amplifier's rise time is defined as the time 
required for the output to rise from 10 to 90 per cent of the final steady- 
state voltage, after the excitation by a unit step. Substituting these 
limits into and solving Eq. (1-29) gives the rise time of a single stage as 

t r = 2.2n (1-32) 

The smaller the time constant, the faster the initial rise rate. The 
rise time ranges from several microseconds in an audio amplifier down 
to a few millimicroseconds in systems designed primarily for pulse 
amplification. 

When the input is a square wave, the amplifier output must eventually 
become periodic. The stray capacity present slows the rise of the leading 
edge, and the coupling capacity introduces some tilt on the flat top. 
The rise time of each half period will be the same as that given by Eq. 
(1-32) for the unit step. However, the amount of tilt depends on the 
duration of the half cycle and must be defined in terms of the wave period. 
If the half period T/2 is small compared with the decay time constant 



Sec. 1-9] 



LINEAB WAVE SHAPING 



19 



r 2 , only a very small decay will occur over each half cycle, and this may 
be approximated by a straight line. From Eq. (1-27), the linear repre- 
sentation of this region, we find that the initial value of the top is B 
(A represents the steady-state value, which in this case is zero). The 
final value, at the end of each half cycle, becomes B(l — Tfbn). Tilt is 
defined as the relative slope of the top of the square wave and is 



8 = 



Ej — Ef 



2t 2 



(1-33) 



S is often expressed as a percentage. Measurement of the tilt for a given 
square-wave input serves as a convenient method of evaluating the hold- 
ing response of an amplifier, just as a measurement of the rise time char- 
acterizes its high-frequency response. 

1-9. Compensated Attenuators. Attenuation of signals through pure 
resistive networks generally proves unsatisfactory because of the stray 
capacity loading of the output (C 2 in Fig. 1-16). Instead of the rapid 
response to a step expected from a resistive attenuator, the output rises 
with a time constant 

t = (7,[(fii + R.) || Ri] 
toward the final value 



E„ 



Rz 



Ri + R e -(- R2 



E 



In order to prevent the source impedance from affecting the signal 
division, usually R x S> R,; otherwise a change from one network driving 
source to another would also result in a new attenuation ratio. 





Fig. 1-16. Uncompensated attenuator. Fig. 1-17. Circuit of the compensated 

attenuator used for the calculation of the 
initial rise. 

The attenuator response can be greatly improved by shunting Ri 
with a small capacitor C\. Since it furnishes a low-impedance path 
to the initial rise, the output will reach steady state much sooner. At the 
instant of closing the switch, the two uncharged capacitors are effectively 
short circuits. Since most of the current will flow through them, rather 
than through the parallel resistors, we shall remove Ri and Ri from the 
circuit while investigating the initial rise (Fig. 1-17). The output now 



20 



MODELS AND SHAPING 



[Chap. 1 



rises with a time constant 



toward 



Tl 



R, 



E,,i = 



Ci + C, 
Cl E 



Ci + d 



(1-34) 
(1-35) 



By assuming that steady state is reached before the decay begins, we 
can draw the model of Fig. 1-18 to represent the final portion of the 
response. In this circuit the initial charge on each capacitor is indicated. 

The output now rises or decays from its initial value toward a steady- 
state value -E ss 2. 

E M = n R * n E (1-36) 



R\ + J?2 



The new time constant becomes 

r 2 = (fii || fl,)(Ci + C) 



(1-37) 



Comparison of Eqs. (1-34) and (1-37) indicates that t 2 is much longer 
than n. C\ and C 2 in parallel are obviously larger than the same two 
capacitors in series. R„ had been assumed small, so that the parallel 
combination of Ri and # 2 will probably be the bigger term. 



rW 



r 



Ci 



e 2 (t) 






-=-£ 



I 



c 2 



E R z $ Cz^^-E e 2 (t) 



Overcompensated 




Fig. 1-18. Decay model — compensated 
attenuator. 



- Undercompensated 
No compensation 



t 

Fig. 1-19. Compensated-attenuator out- 
put (slightly exaggerated for purposes of 
illustration). 



If the two steady-state values [Eqs. (1-35) and (1-36)] are equal, 
there will only be one transient, the initial rise. By equating these two 
voltages and solving, we arrive at the conditions for the optimum attenu- 
ator response, 

R\C\ = R2C2 (1-38) 

Compensating the attenuator greatly improves the rise time, as shown 
in Fig. 1-19. When .R1C1 < jR 2 C 2 the attenuator is overcompensated, 



LINEAR WAVE SHAPING 



21 



introducing an overshoot. If we are interested in attenuation of triggers 
(sharp pulses), the overcompensated case is often desirable since it gives 
the fastest rise and the largest initial amplitude. Much to our regret, 
the .RC-coupled amplifier cannot be compensated because r p does .not in 
fact exist as an entity across which a capacitor can be connected. Any 
external capacity added only parallels the existing strays. This simply 
increases the rise time, with a corresponding degeneration in the over-all 
response. 

The initial rise time of the compensated attenuator is usually so short 
that it may be ignored. This leads to the assumption that the output 
instantaneously jumps to the point from which it starts decaying toward 
the final steady-state value. 

Treating the same problem from the pole and zero viewpoint and 
neglecting the initial rise (R, = 0), we see that the placing of Ci across 
Ri introduces a zero along the real axis. Proper adjustment moves 
it into coincidence with the pole determined by R2C2, annihilating that 
pole. Since the circuit no longer contains any poles, the response cannot 
be of exponential form but must be constant. 



PROBLEMS 

1-1. The switch in the circuit of Fig. 1-20 is closed at t = and reopened at t = 10 
msec. 

(a) Sketch the voltage waveshape appearing across the capacitor, giving the values 
of all time constants and break voltages. 

(6) Sketch the waveshape of the current flowing through the switch and evaluate 
the initial and final values of this current by using the information of part a. Do not 
solve the exponential-response equation in this part. 



50vS: 



2K IK 



•2K 



Fia. 1-20 



=t=2/*f 



100 v-^ 




Fig. 1-21 



1-2. In Fig. 1-21, Si is closed at t - and <S 2 1 msec later. Evaluate and sketch 
on the same axis the current response of both inductances. Make whatever reason- 
able approximations are necessary to simplify the calculations, including changing 
the ideal elements to not-so-ideal elements which become ideal in the limit. Repeat 
if Si is closed 100 j*sec after Si. 

1-3. The capacitor of Fig. 1-22 is initially charged to 50 volts with the polarity 
shown. The circuit is energized by closing Si at t = 0. When the output reaches 



22 



MODELS AND SHAPING 



[Chap. 1 




— 10 volts, switch Si is closed. Draw the output voltage waveshape, indicating all 
time constants, and solve for the time delay between the closure of the two switches. 



100 v-= 



Fig. 1-22 

1-4. Repeat Prob. 1-3 if the bottom of the 1-megohm resistor is returned to +200 
volts with respect to ground instead of directly to ground. Si will now be closed when 
the output rises to zero. All other conditions remain unchanged. 

1-6. A rectangular voltage wave, such as the one shown in Fig. 1-3, is the driving 
signal applied to a series RC circuit having a time constant of 10 msec. The 10-volt 
positive peak lasts for 30 msec, and the — 2-volt negative peak for only 5 msec. 

(a) Find the steady-state minimum and maximum voltages appearing across the 
capacitor. 

(6) What voltage would be read on a d-c voltmeter connected across the capacitor? 
Explain your answer. 

1-6. The excitation of a parallel RL circuit is a square wave of current having a 
peak-to-peak amplitude of 50 ma and a period of 200 Msec. The inductance is 1 henry, 
and the resistance 5 K. 

(a) Sketch the steady-state node voltage and evaluate the maximum and minimum 
values. 

(6) Find the power dissipated in the resistance. 

1-7. The periodic current flowing in the series RLC circuit of Fig. 1-23 is given. 
What waveshape and values of excitation voltage must be applied to cause this cur- 
rent flow? 

<1 
+ 100 ma 



-100 ma 

Fig. 1-23 

1-8. A periodic triangular voltage of 200 volts peak to peak appears across the 
parallel RLC circuit shown in Fig. 1-24. 



/ 1 
/ 1 
f 1 


l\ 

1 \ 

I 


\30 40 


50 


/ 1 t,jUSQC 


10 


20 


\ \ 
\ i 


i 
i y 


/60 70 * 













Fig. 1-24 



LINEAR WAVE SHAPING 



23 



(a) Calculate the inductance that will make the peak current supplied equal to 
100 ma. (The d-c level of the input current is zero.) 

(b) Sketch the current waveshape and give the values at all break points. 

1-9. The nonperiodic signal shown in Fig. 1-25 is applied to a series RC circuit. 
Calculate the approximate response under the following conditions: 
(a) The voltage across C when the time constant is 10 msec. 
(6) The voltage across R when the time constant is 10 msec. 

(c) The voltage across R when the time constant is reduced to 10 jjsec. 




t.jisec 



Fig. 1-25 



1-10. (o) The triangular voltage of Fig. 1-24 is applied to an integrator circuit 
having a time constant of 25 msec. Sketch the steady-state voltage appearing 
across C. 

(b) If a signal consisting of only the positive portion of the triangular wave of 
Fig. 1-24 were applied to this integrator, how long would it take for the output to 
rise to 9 volts? 

1-11. (a) Under what conditions will one of the branch currents of a parallel RC 
circuit be proportional to the integral of the driving signal? Which branch? 

(6) Show that a parallel RC circuit can also be used for current differentiation. 

1-12. Repeat Prob. 1-1 1 with respect to a parallel RL circuit. Explain what further 
restrictions, if any, must be imposed as a result of using a nonideal inductance (wind- 
ing resistance) . 

1-13. (a) A 100-volt symmetrical square wave having a period of 100 iiaee is applied 
as the input to an integrator having a time constant of 10 msec. The output is 
coupled through an ideal isolation amplifier which multiplies the voltage by a factor 
of 50. The amplifier output is then passed through a differentiator having a time 
constant of 1 /jsec. Sketch the ultimate output. 

(6) The same circuits and input are again used except that the square wave is first 
differentiated, then integrated, and finally amplified. Sketch the output appearing 
in this case and explain any discrepancy between this answer and the one for part a. 

1-14. The periodic waveshape of Fig. 1-6 is used as the input to the circuit of 
Fig. 1-5. 

(o) Sketch and evaluate the output waveshape under the following conditions: 
t = 400 jisec, t = 200 msec. 

(6) Drawing on the answers of part a, sketch, without evaluating, the output wave- 
shape appearing when t = 20 msec. Justify your answer without finding the exact 
solution. 

1-16. Two signals are combined in the summation circuit of Fig. 1-26. Draw the 
output, giving voltage values under the following conditions: 



24 



MODELS AND SHAPING 



[Chap. 1 



(a) The circuit is as shown below. 

(b) We replace the 10-K resistor by a 0.05-juf capacitor. 

(c) The circuit is further modified by also replacing the 500-K resistor with a 
0.005-juf capacitor. 



100 v 

t 
c : 

-100 



100 v 















? 3 




6 7 


1 




4 5 




t, 

msec 












1 M 



500 K 
e 2 o WW- 



.10 K e„ 



Fig. 1-26 

1-16. We are interested in generating a reasonable approximation to the signal 
shown in Fig. 1-27. Under the conditions listed, find the circuit required and the 
values of components needed. Make your smallest resistor 10,000 ohms. 

(a) The only signal available is ei of Fig. 1-26. 

(b) The only signal available is a 200-volt peak-to-peak square wave of the proper 
period. 

(c) Repeat parts o and b but use a series combination of circuit elements for the 
output branch instead of a summation-type circuit. 

1-17. A single-stage amplifier having the equivalent circuit shown in Fig. 1-13 is 
excited by a 1-volt step at the grid. The tube and circuit components are r v = 10 K, 
n = 25, R L = 20 K, R, = 100 K, C, = 200 n&, and C = 0.01 fit. 

(a) Find the output response, giving all time constants and steady-state values, 
by the approximation method discussed in the text (time yourself). 

(6) Solve for the exact response, using either classical or transformation methods, 
and compare both the time required and the accuracy of solution with those found in 
part o. 




1-18. In the circuit of Fig. 1-28, a unit step of current is applied at t =0. Find t'i. 
Sketch your answer and give all time constants and steady-state values. Justify the 
approximations made in the course of your solution. 



LINEAR WAVE SHAPING 



25 



2.2 K 



T~7>\ 



'100 



0.5 h 



100 mh 



Fig. 1-28 

1-19. (<z) Derive Eqs. (1-30) and (1-33). 

(6) Calculate the rise time of the amplifier of Prob. 1-17. 

(c) What is the lowest-frequency square wave that would be transmitted without 
introducing more than 5 per cent tilt? Sketch the output if the peak-to-peak voltage 
is 10 volts. Give the actual voltage values. 

(d) In order for a pulse to register its presence at some later point in the system, 
at least 80 per cent of it must be transmitted by the amplifier of Prob. 1-17. What is 
the narrowest pulse that may be used, and how will it appear after passing through 
the amplifier? 

1-20. Find the output response of the circuit of Fig. 1-29 if the input is a 100-volt 
peak-to-peak square wave having a period of 200 /isec and if Li is (a) 100 mh, (6) 
500 mh, (c) 400 mh. Plot the three waveshapes to scale on the same graph. 

5M 
Xo WA, 



o — '000' 1 VW ' 

Lj 20 K 


i o 




>5K 


«in 


k e 


100 mh< 





yo- 



1 M 



100 /^f 



>10K 4:0.01 
Mi 



Fig. 1-29 



Fig. 1-30 



1-21. The two input signals of Fig. 1-26 are applied to the circuit of Fig. 1-30. 

(a) Sketch the steady-state output, giving all voltage values when ei is connected 
to terminal X and ej to terminal Y. 

(b) Repeat part a if a 20-M/»f capacitor is inserted across the 5-megohm resistor, 
(e) Repeat part 6 with the two signals interchanged. 

BIBLIOGRAPHY 



Brenner, E., and M. Javid: "Analysis of Electric Circuits," McGraw-Hill Book Com- 
pany, Inc., New York, 1959. 

Guillemin, E. A.: "Introductory Circuit Theory," John Wiley & Sons, Inc., New 
York, 1953. 



CHAPTER 2 

DIODE WAVE-SHAPING TECHNIQUES 



Wave-shaping functions performed by purely linear circuits — differ- 
entiation, integration, summation, and not much else — are relatively 
limited. If at some point within any operation we could, at will, change 
the value of even one circuit component, there would result a tremendous 
increase in the possible shapes of the output produced. For example, 
if in the middle of integrating a signal, the time constant were suddenly 
to be greatly decreased, then the circuit would immediately become a 
differentiator, with a resultant total output that could not possibly be 
produced in a purely linear circuit. When considering periodic inputs, 
the component changes should be voltage- or current-controlled, e.g., a 
bivalued resistor — a high resistance at voltages below a threshold value 
and low resistance thereafter, or vice versa. Otherwise, the circuit 
change would not occur at the same voltage point of each cycle and a 
periodic output would not result from the periodic input. One method of 
obtaining this response is through the use of two resistors and a switch, 
which is thrown from one to the other and back again as the circuit 
voltage rises or drops past the threshold value. Of course, for fast 
complex waveshapes, mechanically operated switches could not possibly 
operate rapidly enough, and how could we throw the switch continuously? 

Conveniently for us, there exist a number of voltage-controlled non- 
linear devices with approximately the properties desired. One of the 
most common of these devices is the diode, and in this chapter we propose 
to examine its application to various circuits, making use of its inherent 
nonlinearity. 

2-1. Passive -nonlinear-circuit Representation. The response of any 
network to a forcing function is determined by the individual character- 
istics of the various components comprising that network. In the 
analysis of so-called linear circuits, ideal elements (resistors, capacitors, 
inductors, and voltage and current sources) are convenient fictions 
introduced to simplify engineering calculations. The actual physical 
elements are not only slightly nonlinear, but also include various para- 
sitics; e.g., a capacitor contains lead and winding inductance and its 
capacity varies somewhat with the charge storage. Analogous char- 

26 



Sec. 2-1] diode wave-shaping techniques 27 

acteristics may be detected in all other "linear" components. These 
minor second-order effects will cause at most a slight correction in the 
calculated ideal-network response. Their inclusion in the problem will, 
however, obscure the phenomena under examination beneath the com- 
plexity of the additional mathematics. Thus, in the interests of sim- 
plicity, we are completely justified in treating only ideal linear elements. 
Following a similar argument, we can postulate the existence of an 
ideal nonlinearity, which we shall call the ideal diode, to account for 
the response of a large class of grossly nonlinear systems. This element 
has as much validity as our other circuit components and will be utilized 
in networks in much the same manner. It has the bivalued character- 
istics expressed both graphically and in equation form in Table 2-1. 
The ideal diode is an absolute short circuit for any positive current flow 
and an open circuit when the polarity across its terminals is negative 
(back-biased). Its volt-ampere plot consists of two connected segments 
with the break between the two regions occurring at e<( = (or id = 0). 
Thus the ideal diode is identical with the voltage-controlled switch pre- 
viously discussed, the switching value being zero volts. 

Table 2-1 



Element 



Symbol 



Defining equations 



Volt-ampere 
characteristic 



Ideal diode. 



Resistor . 



Voltage source . 



Current source. 




<> 



ed = or id > 
and ed < or id = 



es =- iR 
i = Oe B 



e, = constant =- E 
or e, = e(t) 



i, = constant = / 
or i, = i(l) 



,-j Slope 
0/G 



Table 2-1 illustrates the schematic representation of the branch ele- 
ments, with which we shall be concerned, together with their defining 
equations and volt-ampere plots. 

The combined circuit response of these elements is found directly 
from Kirchhoff's voltage and current laws. Any number of elements in 
series must have the same current flow, and thus the total terminal 



28 



MODELS AND SHAPING 



[Chap. 2 



voltage will be the sum of the individual drops measured at this current. 
A parallel arrangement has an input current that is the sum of the 
current flow in each branch evaluated at the common terminal voltage. 
The diode introduces constraints dictated by its nonlinearity. 



«r<« 



.X 



Q\> 



l a 


|e« 






e d / 




/\ 


J 


/<* 


V 


•'Slope 5 


/ 





IE 


e 


/ 
/ 









(a) 





I 


7 


/ 
/ 
/ 
/ 
s 


I'* 



(b) 

Fig. 2-1. Two biased diode circuits and their volt-ampere characteristics, (a) Series 
combination of elements; (b) parallel combination of elements. 



Let us now consider the two examples shown below. In Fig. 2-la 
the series diode limits the current flow to positive values, the battery 
shifts the break point to the right, and the resistor introduces a finite 
slope in the conduction region. Summing the individual equations given 
in Table 2-1 leads directly to the defining equations for the compound 
branch. 



c« = e d + e B + e. 




e„i = E + i a R for i a > or e„ > E 


(2-la) 


e„2 = ed + E for i a = or e„ < E 


(2-16) 



Two equations are necessary because of the bivalued diode properties: 
one holds for the conduction region (2-la), and the other for the back- 
biased zone (2-16). They agree at the boundary point, which may be 
expressed in terms of the applied terminal voltage by setting i = in 
Eq. (2-la) and ea = in Eq. (2-16). Figure 2-la illustrates the graphical 
summation of the individual curves, i.e., the superposition of the element 
voltage drops. We might note that the voltage drops add normally 
in the positive current region but that the reverse-biased diode (ed < 0) 
absorbs the total loop voltage. 



Sec. 2-1] 



DIODE WAVE-SHAPING TECHNIQUES 



29 



Reversal of the diode would establish the conduction zone for negative 
rather than for positive currents, and reversal of the battery would 
shift the break point to the left instead of to the right. The diode 
connection determines the permissible current-flow direction, and the 
battery voltage only sets the break-point location. 

In Fig. 2-16 the currents must be summed over the voltage range per- 
mitted by the diode (for negative voltages it is a short circuit forcing 

ia = to + id + i, 

iai = Ge a + I e„ > or i a > I (2-2a) 

i«2 = id + I e„ = or i„ < I (2-26) 

The shunt current source shifts the current coordinate of the break 
point to a positive value as shown in the graphical characteristic of Fig. 
2-16. Again, the diode restricts the region where graphical addition is 
necessary, in this case by introducing a short circuit upon conduction. 






1 
1 

Slope Gi i 




'a 

Slope fG 1 +G 2 ; 
/ 


1 










-E 





e o 



(a) (b) 

Fia. 2-2. (a) Combined biasing of piecewise-linear circuit; (6) volt-ampere charac- 
teristic. 

Even if these circuits did not contain a nonlinear element, the inter- 
cepts of the series-resistance and parallel-conductance curves would still 
differ from zero by the value of the voltage and current biases, respec- 
tively. The addition of the diode only limits the portion of the total 
shifted curve which may be used. 

In general, we can say that : 

1. If there is no current source included across the diode, then one 
coordinate of the branch break point will always be the bias voltage con- 
tained in the diode branch. 

2. If there is no voltage source included in a parallel arrangement of 
elements or branches, then one coordinate of the break point will always 
be the value of the current bias across the diode. 

Combining parallel current and series voltage biasing with the single 
diode permits us to locate the break point anywhere in the volt-ampere 
plane. Moreover, by including both series and parallel padding resist- 
ance, the slopes in the two zones may be individually controlled (Fig. 2-2). 



30 



MODELS AND SHAPING 



[Chap. 2 



Suppose that we are interested in evaluating the input volt-ampere 
characteristic of any single diode circuit. How would we do so? We 
could, of course, solve the problem graphically, by summing the response 
of each element. For more complex circuits, such as the one shown in 
Fig. 2-2, this process may become somewhat tedious. 



& 



±_ fl,E 



-B.+Jfe 



(a) 



fi 2 




l< 1 



-X* 



-E 



fb) 



i°i 



)\i 





I 




°'°P e r,i\rJ 

<s^> — / 




^- -E - R i E 

S l +R ! 





e« 



(c) 

Fio. 2-3. Models illustrating the solution of the circuit of Fig. 2-2. (a) Equivalent 
circuit when diode conducts and the Thevenin equivalent circuit; (fc) circuit when 
diode is back-biased and the Norton equivalent circuit; (c) composite volt-ampere 
characteristic. 

For an algebraic solution, the simple diode circuit may be replaced 
by the two implicit resistive models, one of which holds when the diode 
conducts and the other when it is back-biased. By superposition from 
the model of Fig. 2-3o, where the current source is short-circuited by the 
conducting diode, 

e " +1- ii>0 (2-3) 



la = 



Ri I Ri Ri 



From the second model (Fig. 2-36), which holds for d < 0, we obtain 
the other denning equation, 



= — + J 
Ri 



(2-4) 



Sec. 2-1] diode wave-shaping techniques 31 

The boundary between the regions is the intercept of the two straight 
lines whose equations are given by (2-3) and (2-4). This is shown 
graphically in Fig. 2-3c. Equating and solving for the terminal break 
value eab yields 

eat, = IRt- E 

The current break coordinate t„j can now be found by substituting e^ 
into either defining equation. 

We might observe that the break does not occur at the intersection of 
i a = I and e a = — E. This is a direct consequence of the additional 
current flow through Ri (due to E) and the additional voltage drop 
across Ri (due to I). The reader might remove each resistor in sequence 
and- observe the effect on the break-point location. 

Each of the two models may be reduced to the Thevenin or Norton 
equivalent circuit, shown in Fig. 2-3, by algebraic manipulation. The 
open-circuit voltage or short-circuit current source is the value of the 
appropriate intercept, and the slope of the line segment is the resistance 
or conductance. 

Since it is known that the volt-ampere curve consists of two linear 
contiguous segments, it follows that three pieces of information are 
sufficient to define these regions. Instead of writing the complete equa- 
tion, we can simply solve for the coordinates of the break point and the 
two adjacent slopes. While doing so, attention must remain focused on 
the diode because its state controls the circuit behavior. Example 2-1 
illustrates this technique for the circuit of Fig. 2-2. 

Example 2-1. If we select the conducting state in Fig. 2-2 as our arbitrary starting 
point, the diode will short the cun ent source and reduce the network to one contain- 
ing two resistors and a voltage source (Fig. 2-3o). This occurs where the diode cur- 
rent is zero and where the current through Rz equals I. Thus the break voltage is 
the total branch drop at this current flow, 

e„i = IR2 — E 

For Ri = 10 K, Rt = 1 K, E = 25 volts, and 7 = 5 ma, 

e„ h = 1 K X 5 ma - 25 volts = -20 volts 

The other coordinate becomes the sum of the current through R it because of the 
applied break voltage e.t and the bias current /. From Eq. (2-4), 

—20 
tat, = 5 ma + „ = 3 ma 

Within each region the slope is found by calculating the incremental conductance: 
first when the diode is conducting (e„ > —20), and next when it is back-biased 
(e„ < —20). These values are already indicated in Fig. 2-3c. In the forward region 
it is 1.1 millimhos, and in the back-biased zone it is only 0.1 millimho. 



32 



MODELS AND SHAPING 



[Chap. 2 



We conclude that once the nonlinear circuit is represented by a model 
containing a diode, we can write two sets of linear equations to describe 
the system response, one equation holding while the diode conducts 
and another when it is back-biased. These' equations must coincide 
at the boundary point (ed = 0); the continuous diode characteristic 
introduces no voltage or current discontinuity, and neither should the 
model or the equation. By using two regions to represent the nonlinear 
device, all theorems and techniques of the solution of linear circuits 
become applicable in each region. The only complexity introduced is 
that upon completion of a solution we must check to see that we have not 

left one linear region and entered 
another. If the circuit has done 
so, then the original equation, 
written for one region, will not hold 
and the answer will be incorrect. 
The problem must subsequently 
be reformulated and re-solved. 

2-2. Physical Diode Model. A 
semiconductor diode has volt- 
ampere characteristics of the type 
shown in Fig. 2-4. Note the scale 
change as voltage and current 
become negative. Vacuum-diode 
characteristics are similar, differing 
primarily in their having a much larger forward voltage drop and positive 
rather than negative reverse current. 

Circuit analysis is greatly simplified once the actual characteristics 
are approximated by the two straight-line segments shown superimposed. 
When drawing these lines, it is convenient to start from the origin and 
intersect the actual characteristics at the center of the operating region. 
However, since this is only an approxi- 
mation and the individual diode devi- 
ates rather widely from manufacturers' 
data, we find any reasonable set of 
lines acceptable for most purposes. 
Their slope has units of conductance. 
We conclude that the diode may 
reasonably be represented by two con- 
stant resistances, a forward resistance 
Tf, when ed > 0, and an inverse resist- 
ance r r , for ed < 0. The change from one to the other takes place 
when the voltage across, and the current through, the diode drops 
to zero. Thus the model formulation makes use of the ideal diode 




Note change 
of scale 



Fig. 2-4. Semiconductor diode character- 
istic. 



Qt 



<fc 



Fig. 2-5. Diode model representa- 
tion. 



Sec. 2-3] 



DIODE WAVE-SHAPING TECHNIQUES 



33 



as the switching element (Fig. 2-5) ; it shunts the very large r r with the 
much smaller »y upon conduction at e<i > 0. Therefore ry and r r in 
parallel are almost exactly equal to ry. If this inequality did not happen 
to hold, in any particular case, we would modify the model by increasing 
the series resistance so that the parallel combination is the actual forward 
resistance of the diode. The capacitor shown in the model represents 
the major parasitic element present. 

Forward-resistance values range from a fraction of an ohm to 250 ohms, 
and reverse resistance from about 25 K to 1 megohm, depending on the 
particular device. The lower values generally appear in power rectifiers, 
and the higher ones in general-purpose and switching diodes. Special- 
purpose diodes are available having very small forward and extremely 
large reverse resistances, and these very closely approximate an ideal 
device. 




E,r^L 



(a) (b) 

Fig. 2-6. Zener-diode characteristic and model. 

Silicon diodes designed for operation into the Zener voltage region 
have the volt-ampere characteristics of Fig. 2-6a. It may be seen that 
three linear regions are necessary to define properly the complete curve, 
and therefore two ideal diodes are required in drawing its model (Fig. 
2-66). Generally, the small Zener-region resistance rz may, as a first 
approximation, be taken as zero. 

2-3. Voltage Clipping Circuits. One of the most widely used nonlinear 
circuits utilizes the bivalued properties of the diode to control the signal 
amplitude. In the ideal shunt clipper of Fig. 2-7a, when the diode is 
back-biased, the shunt branch is opened and the input voltage is trans- 
mitted through Ri to the output, unchanged and unaffected by the 
network. 

On the other hand, once the rising input forces the diode into con- 
duction, it may be replaced by a short circuit. The subsequent output 
is E, provided, however, that it is exceeded by the input. Thus the 
two zones of operation are, first, a transmission region where the output 
equals the input (e„i = e ln ) and, next, a clipped region where the output 



34 



MODELS AND SHAPING 



[Chap. 2 



is determined solely by the bias (e.,2 = E). These zones are plotted as 
the ideal transfer characteristic in Fig. 2-8. 

Turning now to the physical diode clipper shown in Fig. 2-76, the 
addition of r r and jy establish information transmission paths which are 
nonexistent in the ideal circuit. In each of the operating regions only 
one of the diode resistors need be considered and the actual output may be 



o-WV — | 

L 


1 

t 




I 1 

+ e < 


_ 


L 



o WW- 



X 



r r >e d 






1- 



(a) (b) 

Fig. 2-7. Voltage clipping circuit (shunt diode), (a) Ideal model; (6) physical model. 




Fig. 2-8. Clipper transfer characteristics. 

found by the linear superposition of the contributions from both voltage 
sources. When c u < E and the diode is back-biased, 



e i = 



r, 



«i 



r r 



e iB + 



Ri 



Ri + r r 



E 



(2-5) 



When conducting (e ln > E), the diode may be replaced by its forward 
resistance and the output voltage may be written 



Co2 



Ri 



Ri + r, 



E + 



r/ 



Ri + r, 



(2-6) 



Both equations, each defining the operation in an individual region, 
are straight lines, as shown in Fig. 2-8. At e in = E, the break voltage, 
both Eqs. (2-5) and (2-6) give the output as E, proving the continuity 
of the transfer function. 



Sec. 2-3] diode wave-shaping techniques 35 

A plot of Eqs. (2-5) and (2-6), the transfer characteristic of the clipper, 
appears in Fig. 2-8, with the ideal characteristic also presented for com- 
parison. Examination of Eq. (2-5) indicates that within the trans- 
mission region, the finite diode reverse resistance results in an undesirable 
contribution to the output from the bias E. This contribution is the 
second term of the equation. In the clipping region, the second term of 
Eq. (2-6) represents the output contribution from the now undesired 
input. We are faced with the necessity of optimizing this circuit, helped 
in this by our knowledge of the behavior of the ideal circuit. 

Freedom of choice is limited to selecting an optimum value for Ri. 
Rewriting Eqs. (2-5) and (2-6) in the form of (2-7) and (2-8) will aid in 
formulating the problem. 



Col = 



ET7,«-( 1 + *3 (2 " 7) 



-■^ J! ( 1+ J:i) (2 - 8) 



Ri 

Ri 

Ri + r, 

The term outside the parentheses in both Eqs. (2-7) and (2-8) consists 
of a constant multiplying the signal producing the desired output infor- 
mation in the several regions. We might consider the term inside the 
parentheses as representing the desired output (unity) plus a proportional 
error term. The error in both equations includes the ratio of the signal 
producing the undesired transmission to the signal producing the desired 
transmission. Equation (2-7) holds when the diode is nonconducting, 
and here this ratio is E/e ia > 1 . In Eq. (2-8) , defining the system response 
for diode conduction, the ratio of e in /E is also greater than or equal to 
unity. Since the complete second terms of both equations represent 
the proportional error, they should, for optimum response, be made small 
compared with unity. The conditions on Ri are contradictory: in 
Eq. (2-7) the optimum value becomes Ri = 0, and in (2-8), Ri = °°. 

We shall define the two error terms in the parentheses as 



«ol 

and £02 



r r VWi 
Ri\Ej 2 



To resolve the contradiction, "a" compromise choice for Ri will be 
designated the optimum value. One possibility is a resistance that 
results in error terms of the same magnitude in both equations when the 
ratio of the signal producing the undesired output to the signal producing 
the desired output is the same, or «<,x = «„ 2 when 



U)i (£)« 



36 MODELS AND SHAPING [CHAP. 2 

Consequently, the compromise value becomes 

Ri = V^r (2-9) 

Substituting Eq. (2-9) back into the proportional-error terms yields 

We conclude that for clipping circuits a diode with the highest ratio 
of inverse to forward resistance represents the best choice. In any 
individual circuit, the series resistance used may not be the one found 
above; particular constraints may dictate a value closer to zero or to 
infinity. 

+ Ri + 

o — AV- 




X 



e in 



^ 



(b) 
Fig. 2-9. Negative clipper transfer characteristic and circuit. 

The use of the Zener region of the diode in addition to the forward- 
conduction region sets two clipping values instead of one. One point 
is located at e d = 0, and the other at e d = E z , as indicated in the volt- 
ampere characteristics (Fig. 2-6). Biasing this diode shifts both break 
points in the same direction while still maintaining their original separa- 
tion. It follows that one difficulty facing us is the lack of individual 
control over the slopes in the two conduction regions. Any series resist- 
ance added contributes equally to the Zener region and the diode forward- 
current region. Generally, for versatility of adjustment, two individual 
diodes are preferred to one Zener diode. However, the constant voltage 
drop in the Zener region does avoid the necessity for inserting a separate 
bias source. 

When the diode in Fig. 2-7 is reversed, the clipping region is below 
the break point; the diode now conducts through its forward resistance 
Tf until e in > E. After conduction ceases, a transmission path through 
the reverse resistance still exists, and the final result is the transfer 
characteristic given in Fig. 2-9. Here the signal is clipped below the 
bias voltage and transmitted above. Reversing the polarity of the bias 



Sec. 2-4] diode wave-shaping techniques 37 

voltage E only shifts the break point of the transfer characteristic to 
the third quadrant. We conclude that the direction of the diode connec- 
tion determines whether clipping will be below or above the bias value and 
that the bias voltage determines the actual clipping point. 

Series clipping, a possible circuit variation, gives results similar to 
those found for the previously discussed shunt clipping. Insertion of the 
diode in series with the transmission path (Fig. 2-10) allows transmission 
while it conducts, in this case for 
Cm < E. When back-biased, the +_ 
diode introduces a high series im- f 
pedance as the means of limiting I , 

the signal output. In this respect i + -1- 

it behaves just opposite to the shunt | -T 



i-H-^Wv- 



clipper, which stops transmission r ^ X - 

upon conduction. The transfer char- ^ 2-10. Series clipper circuit. 

acteristic of the circuit of Fig. 2-10 

is of nearly identical shape with that of the shunt clipper of Fig. 2-7 
(differing in the slopes), and the optimum value of Ri will also be the 
same as that discussed above. 

2-4. Single Diode and Associated Energy-storage Element — Some 
General Remarks. In the solution of circuits containing both ideal 
diodes and energy-storage elements, notice must always be taken of the 
immediate past history as well as of the present excitation. The storage 
element should be considered as a memory which contains information 
stored in the past and which permits history to play a role in determining 
the present and future state of the circuit. 

Our mode of attack follows the principle of time-zone superposition 
developed in Chap. 1. First the initial state of the circuit is determined; 
next the network is solved for the time at which the diode will change 
state. 

Thereafter we simply treat the new circuit with the new time constant 
and initial conditions. The energy-storage element adds constraints to 
the continuity of current or voltage across the circuit and/or excitation 
changes and in this manner contributes to the new initial values. 

Example 2-2. We shall now consider the problem posed in Fig. 2-11, where the" 
initial voltage across the capacitor is 100 volts (its polarity is indicated in the sketch) 
and where the negative 50-volt 1-msec pulse is applied at t = 0. The capacitor 
charge remains constant across the instant of pulse application, and thus the output 
drops from —100 to —ISO volts. Since the diode remains back-biased, the voltage 
immediately starts rising toward +300 volts with a time constant of ti = 1 msec 
(0.01 pi X 100 K). The defining equation becomes 

e.i = 300 - 450e-" r i 



38 



MODELS AND SHAPING 



[Chap. 2 



■50- 



0.01 fd 




Fig. 2-11. Diode and capacitor circuit for Example 2-2, showing an input and the 
resultant output waveshape. 

Eventually the output reaches zero, the diode conducts, and the circuit changes. 
Solving the above equation for the time at which e„i = 0, 

(, = ti In 45 %oo - 0.405 msec 

In the new circuit the output voltage will charge toward the TheVenin equivalent 
value of 

j^ 300 -6 volts 



Eth — 



with a new time constant t% = 20 ^sec (2,000 ohms X 0.01 ^f). We can assume 
that the circuit reaches this value in 80 /usee and that it will remain there until the 
excitation again changes. 

At the termination of the pulse (I — 1 msec), the input rises by 50 volts and this 
same jump is coupled by C to the output. Since this does not change the diode state, 
the output recovers to 6 volts with the short conduction time constant of 20 /xsec. 
The complete waveshape is shown in Fig. 2-11. 

If a capacitor were placed across the diode, then the charge stored in it 
would maintain the diode state invariant for some time after the excita- 
tion change. An inductance in series would produce a similar effect by 
sustaining the current flow. We conclude that the location as well as 
the type of energy-storage element employed directly influences the non- 
linear behavior of a single-diode system. In any analysis great care 

should be taken to see that the con- 
trolling factor, the diode state, is 
always kept in view. 

Some practical widely used circuits 
employing various energy-storage ele- 
ments will be discussed in detail in 
the sections below (2-5, 2-7, and 2-9), 
and they will further illustrate the 
modes of solution. 

2-5. Diode Voltage Clamps. The 

function of the diode clamp of Fig. 2-12 

is to shift the input voltage in such a manner that the resultant 

maximum value of the output will be maintained at zero, without at 




Fig. 2-12. Diode voltage clamp. 



DIODE WAVE-SHAPING TECHNIQUES 



39 



Sec. 2-5] 

the same time distorting the waveshape. The output voltage is then 
said to be clamped negatively at zero. Sketches of typical input and 
output waves appear in Fig. 2-13. 



E, 



cm 



<i 



E, 



-E 





h h 


■ 












«2i* 


+-T 2 -+ 


E 

1 










t 


i 

i 
i 


i 
i 
i 




^ 


-T-r,C 


1 < ' 
> 1 

! J I 




i 

i i 

i 


l 

i 








t 
















t 



Fig. 2-13. Ideal-diode clamping waveshapes. 



In the initial discussion we shall idealize the clamp at least to the 
extent of assuming that r r = °o . Upon application of the input signal, 
C charges to the peak positive value of the input with the time constant 
CV/. Provided that this time constant is sufficiently short, charging will 
be completed within the first cycle; otherwise it may take several cycles. 
Upon completion of the charging, the voltage across the diode becomes 



e d = 



= e ia - E e = e in - #i 



(2-10) 



The conclusion drawn from Eq. (2-10) is that the output will always 
be less than or equal to zero since the voltage across the capacitor, E\, is 
the maximum positive value of e in . Diode conduction ceases once C 
is fully charged with the diode subsequently remaining back-biased. 
The output never exceeds zero, and it is said to be clamped there. For a 
rectangular-wave input, the clamping action is illustrated in Fig. 2-13 
The diode serves first to introduce a low-resistance path for charging, 
and then a very-high-resistance one, preventing the discharge of C. 
Since the output is simply the sum of the capacitor and the input voltage, 
the signal-level shift in the clamp is provided through the charge accumu- 
lated in the capacitor during the initial build-up. The behavior is the 
same as if a battery whose voltage is dependent on the peak positive 
value of the input replaces the capacitor in the series transmission path. 



40 



MODELS AND SHAPING 



[Chap. 2 



During the interval that the diode is back-biased, C discharges slightly 
through the finite value of r r with the long time constant Cr T . The 
energy dissipated in r r is supplied from the charge previously stored, and 
of course it must be replaced once the diode again conducts. The dis- 
charge introduces tilt over the negative portion of the cycle. Since 
the full change of input voltage appears across the diode (because of 
continuity of charge in C), the output becomes slightly positive when the 
input voltage rises by E at the end of each cycle. Therefore the output 
appears as shown in Fig. 2-14. Note that the signal has become some- 
what distorted because of the cyclical charging and discharging. 



Ti=r,C 




- T 2 =r r C 
Fig. 2-14. Physical-diode clamping waveshape. 

If the recharge time constant is excessively long, the circuit will not 
recover to zero within the recharge period. A solution for this case 
would follow the discussion of Sec. 1-3, with, however, different time 
constants used over each portion of the response. 

Some general conclusions as to the circuit behavior may be drawn by 
realizing that for a periodic solution, the net change in the charge stored 
in C over any cycle must be zero. The charge flowing into the capacitor is 



Q+ - f Tl i dt = [ Ti ?± 
Jo Jo r, 



dt 



(2-11) 



where e+ is the time-varying voltage across the output when the diode 
conducts. The charge flowing out of the capacitor is 



Jh r r Jo r r 



dt 



(2-12) 



where e_ is the voltage across the output when the diode does not conduct. 
Since there can be no net change in charge over the cycle, 

Q- = Q+ 

Equating (2-11) and (2-12) and solving for r f /r r yields 



f T ' dt 
r t — J° 6+ — net positive area 

r r f T * e _ fa net negative area 
Jo 



(2-13) 



Sec. 2-5] 



DIODE WAVE-SHAPING TECHNIQUES 



41 



Equation (2-13) leads to the conclusion that for proper clamping as well 
as for clipping, the diode with the maximum ratio of reverse to forward 
resistance performs best (least distortion). Observe that in deriving 
this equation the only assumption made is that the input signal is periodic. 
Even though the waveshape used to illustrate the clamp's behavior is 
rectangular, its shape appears neither explicitly nor implicitly in the 
charge equations (2-11) and (2-12). The final result, that the ratio of 
areas equals the ratio of diode resistances, must be independent of the 
applied waveshape. 

C 
— It— 




X 



iov:=:£ 




Fig. 2-15. A triangular wave clamped positively at —10 volts, illustrating distortion. 

Reversing the diode clamps the input signal positively, instead of 
negatively, at zero volts, with the capacitor recharge showing itself as a 
slight negative excursion. Biasing this clamp similar to the clipping 
circuit sets the output voltage at any desired level, diode direction 
determining positive or negative clamping. Figure 2-15 shows a tri- 
angular input clamped positively at — 10 volts, with the distortion exag- 
gerated for purposes of illustration. 

One necessary condition for properly biased operation is the presence 
of the reverse diode resistance, which establishes a path for the initial 
charging of C to the bias voltage. Consider the behavior when using an 
ideal diode (r r = °o) in the circuit of Fig. 2-15. If the negative peak 
of the input voltage never reaches — 10 volts, the diode always remains 
back-biased and, since C is uncharged, the input appears at the output, 
unchanged and undamped. Signals with negative peaks greater than 



42 



MODELS AND SHAPING 



[Chap. 2 



— 10 volts forward-bias the diode on the peaks, and it provides the 
charging path. However, once the diode has a finite value of inverse 
resistance, C charges through this, and as a consequence the clamping 
action becomes independent of the signal amplitude. 

Referring back to Eq. (2-13), we see that for optimum clamping with 
any given diode, the input signal should be clamped so as to minimize 
the net area on either side of the clamping bias voltage. When the area 
where the diode is back-biased is large, then the distortion area during 
conduction will also be large. 



15 

i° 
-15 



0.5 



5 msec 



(a) 




Fig. 2-16. Clamping waveshapes, (a) Input waveshape; (b) output clamped nega- 
tively at zero; (c) output clamped positively at —30 volts. 

The waveshape of Fig. 2-16a must be clamped so that its maximum 
positive excursion is zero. It may either be clamped negatively at zero 
(output in Fig. 2-166) or positively at -30 volts (output in Fig. 2-16c). 
The diagonally shaded areas in Fig. 2-166 and c indicate the regions 
where the diode is back-biased, and the solidly shaded areas, the regions 
of diode conduction. Since, from Eq. (2-13), the ratio of areas in the two 
cases is the same, clamping the longest portion of the period, rather 
than the shortest, is preferable. This corresponds to the case shown in 
Fig. 2-16c, which has minimum areas and hence least distortion. But 
if the signal amplitude varies, then no choice exists within the constraints 
imposed (maximum positive excursion limited to zero) and the signal 
must be clamped negatively at zero (Fig. 2-166). 



Sec. 2-5] diode wave-shaping techniques 43 

Example 2-3. Suppose that in the clamp used with the waveshapes of Fig. 2-1 6a, 
r/ = 100 ohms, r, = 1 megohm, and C = 0.1 id- The two time constants of interest 
axe 

ti = r r C = 0.1 sec and n - r t C = 10 /jsec 

Since ri is so very much longer than either portion of the cycle, within the back-biased 
region the circuit may be treated a3 an integrator. The output thus decays linearly 
toward the clamping level E. 

(a) When the signal is clamped negatively at zero (Fig. 2-166), the change in the 
output voltage over the 5-msec interval becomes 

Ae 0l =^5X 10-» = 1.5 volts 

At the end of this period the output has risen from -30 to —28.5 volts. The input 
now jumps by 30 volts, driving the output to a positive peak of 1.5 volts, from which 
it recovers with the fast time constant t 2 . 

(b) If the output is clamped positively at —30 volts (Fig. 2-16c), then, within the 
interval where the diode is back-biased (0.5 msec), the output decays by only 

30 
Ae„ 2 = ^j 0.5 X 10-' = 0.15 volt 

Thus during the recharging time the maximum excursion below the — 30-volt level 
is 0.15 volt. The difference between these two cases is directly proportional to the 
duration of the nonconducting portions of the cycle, and we have verified our original 
argument. 

Looking into a clamp, the signal source sees a different complex 
impedance during diode conduction than when reverse diode current 
flows. If the source has any internal impedance, the loading by the 
forward resistance of the diode introduces additional waveshape dis- 
tortion. The total recharge voltage excursion divides proportionately 
across the source impedance and the diode forward resistance. Since 
the output is taken only across r f , the voltage in this region will be smaller 
than expected, as seen in Fig. 2-15. During recharge, the forward voltage 
drop at the output, measured from the bias value, becomes 

E+ is the peak voltage drop across the total series resistance while the 
diode conducts. 

In addition, when using Eq. (2-13), the source resistance must be 
added to the diode forward resistance and may even be the predominant 
term. We can see that this large increase in resistance decreases the ratio 
of areas, with correspondingly poorer clamping. When the diode is 
back-biased, the output is also attenuated slightly because of R,. But if 
R, were large enough to have an appreciable effect, it would so distort the 
conduction signal that the clamper would be unusable. 



44 



MODELS AND SHAPING 



[Chap. 2 



For clamped triangular and sinusoidal signals (Figs. 2-15 and 2-17), 
we are often interested in knowing the diode conduction period. Because 
the source impedance of the input generator only causes division of the 
recharge voltage drop, a logical first step in the calculation is to lump 
all forward resistors and ignore the voltage division. Furthermore, in 
a close-to-ideal circuit the waveshape distortion also may be neglected: 
the clamp is treated as a device which only shifts the output waveshape 
to a point where it satisfies Eq. (2-13). The two areas can now be calcu- 
lated in terms of the unknown conduction angle, and their ratio equated 




Fig. 2-17. A negatively clamped sine wave. 

to the known ratio of forward to reverse resistances. For the triangular 
wave of Fig. 2-15, neglecting distortion, the area above the clamping 
voltage is 

Ax = HTx(Ex + 10) 



(2-14) 
(2-15) 



The area below, measured from the bias voltage, becomes 

Ai = y^T^Ei + 10| 
But from the input wave 

Ex - Et = 40 Tx + T 2 = T (2-16) 

In addition, from the geometry, by setting up the ratio of similar triangles, 

Tx/2 T/2 _ T 2 /2 



T/2 
40 



Ex + 10 



T/2 
40 



\E, + 10| 



(2-17) 



Substituting first Eqs. (2-17) and then (2-16) into Eqs. (2-14) and (2-15) 
and equating to the ratio of resistance, 



r t + R. 



r r 



2V 
TV 



(t - T*y 



(2-18) 



T 2 is the only unknown, and this equation is readily solved. We reject 
the negative root because it has no physical meaning. Next E2 may be 
found by substituting back into Eq. (2-17). Using the resistive voltage 



Sec. 2-6] 



DIODE WAVE-SHAPING TECHNIQUES 



45 



Positive 
clamping 
slope=l 




Negative 
clamping 



Fig. 2-18. Clamper transfer characteris- 
tics. 



divider of R. and r f and recognizing that E t is developed across both 
resistors finally gives E' t . 

Maintaining the clamping level in the face of changes of input-signal 
level requires fast charging of C for increasing signals and fast discharge 
on decreasing inputs. Both conditions are satisfied only with a small 
capacitor. However, too small a value allows excessive discharge during 
the clamp cycle with resultant waveshape distortion. 

Occasionally, clamps are used in systems (such as television) where 
they must respond to the periodic signal but not to any noise pulse which 
might also be present. Since we do not wish the output to shift appreci- 
ably during the narrow noise-pulse 
interval, we must choose a large 
capacity and accept the conse- 
quences of a slow circuit response 
and incomplete recovery within the 
diode conduction region. Or, as an 
alternative, the reverse resistance 
might be reduced so that the effects 
of the noise pulse will be rapidly 
dissipated in the relatively in- 
efficient clamp. In a practical 
circuit a compromise value of r r 

and C would be chosen, satisfying as far as possible all requirements, 
weighing them in order of importance. 

The transfer characteristic of a clamper is given in Fig. 2-18. Note 
that it is a straight fine of unity slope, with the locus of the end point 
the bias voltage line. The y intercept depends on the input-signal 
amplitude. This is obvious if we consider that for positive clamping 
the output must always (neglecting the capacitor recharge) be equal to or 
greater than 2J M „. AEi is the peak negative voltage applied to a positive 
clamper, and AEz is the peak positive voltage applied to a negative 
clamper. These transfer characteristics have no value or meaning 
except for periodic inputs. When presented on an oscilloscope, they do, 
however, show whether the clamp is operating properly. 

2-6. Current Clippers. We find the concept of duality extremely help- 
ful in developing the circuits used for current clamping and clipping. 
By simply taking the dual of a voltage clipper, we arrive at the circuit 
of a current clipper (Fig. 2-19a). Just as loop equations were ideally 
suited for the analysis of voltage circuits, the node equations are ideal for 
treating the current-activated devices. To complete the dual relation- 
ships, where previously the break point of the diode piece wise-linear 
model was taken at e d = 0, now it might well be regarded as occurring 
when id = 0. Figure 2-196 shows the transfer characteristics of the 



46 MODELS AND SHAPING [CHAP. 2 

current clipper. Except for the change in axis (current instead of volt- 
age), it is identical with that given in Fig. 2-9 for the voltage clipper. 
Introduction of a bias current I b sets the break point of the transfer 
characteristic wherever desired. 




(a) (b) 

Fia. 2-19. Current clipper and transfer characteristics. 

Analysis of this circuit starts with the writing of the node equation 

id = ii — It ~ is (2-19) 

When id > 0, the diode conducts. From Eq. (2-19) we see that the con- 
duction region corresponds to ii > I b and the nonconduction region to 
i\ < lb. The break point turns out to be, as expected, the value of the 
bias current. Since the regions of conduction and nonconduction are 
known, the two equations defining operation are, by superposition, 



lol — 
io2 = 



h + 



R 



R + r r *° • R + r r 

R . r, 

R + r/ 1 "+" R + r, 



(2-20) 
(2-21) 



Equation (2-20) holds when the diode is nonconducting, and (2-21) when 
it conducts. The first term on the right-hand side represents the desired 
transmission, and the second term, the error component of load current, 
i.e., the transmission of undesired current to the output. These equa- 
tions are analogous to the set used in describing the operation of a voltage 
clipper [Eqs. (2-5) and (2-6)]. Therefore identical reasoning leads to the 
same optimum value of the shunt resistor R as that previously found for 
Ri in Sec. 2-3. 

R = -\/»V7 

Reversing the diode changes the clipping characteristics, now allowing 
conduction for i\ < I b and clipping when ii > I b . 

2-7. Current Clamps. In the treatment of voltage clamps (Sec. 2-5) 
we saw that the prime constituents of a clamp are an energy-storage 
element supplying the additional energy (previously stored) necessary 
in shifting the signal level, and a diode providing short storage (charge) 



Sec. 2-7] 



DIODE WAVE-SHAPING TECHNIQUES 



47 



and long decay time constants. When we must clamp current wave- 
shapes, we logically turn to the inductance as our storage element. This 
circuit, the dual of the voltage clamp of Fig. 2-12, appears in Fig. 2-20. 
Since its basic behavior is independent of waveshape, the excitation 
chosen might as well be the simplest, e.g., 
a current square wave. For the initial 
discussion, assume its application at 
t = as shown in Fig. 2-21. 

The load current may, by writing the 
node equation, be expressed as 




L-WwvvJ 



to 



id 



ti 



tL 



(2-22) 

Fig. 2-20. Current clamp. 

During the first half cycle of the input 

square wave, the diode conducts and, assuming r f = 0, the full input 

current flows in the output lead. However, in the second half cycle, 

ii < and forward conduction ceases. The input current must flow 

somewhere. Since there was zero initial current in L, the only place left 

for the current to flow is through the diode's inverse resistance. Inductive 

current starts building up toward 

— /„ with a corresponding decay in 

the reverse diode current. Build-up 

is rapid because of the small time 

constant, 

L 

Tl = — 

r r 

For a sufficiently long half period 
compared with n, that is, T/2 > 4t\, 
charging will be completed within the 
half cycle, otherwise within several 
cycles. The input again becomes 
positive and, with no circuit dissipa- 
tion, the inductive current remains 
at -I m . From Eq. (2-22), the out- 
put subsequently becomes 

to = i\ — ( — Im) 

But since t'i takes on only one of two 
values, ± 7„, i„ may be either zero 
or positive but never negative. The output current is composed of two 
components. On the positive peak the contribution of I m , from the 
signal source, and the additional contribution of I m , previously stored 
in the inductance, add. On the negative peaks these components sub- 



Im 










i — . 






H 
-Im 




























t 


1 








l> 


1 

1 
1 














t 


-Im 


l 


--rrrrs — y*-^ 






2/m 


I l 

1 
l 


TZ 










\Im 

















-Im 


















Y 




K 




Y 


t 



Fig. 2-21. Current-clamp waveshapes. 



48 MODELS AND SHAPING [CHAP. 2 

tract. We have thus clamped the output current positively at zero; it 
varies between zero and 27„. 

The inductance acts as the circuit's memory, remembering always 
the peak value of the input excitation. Once this information has been 
stored, the output will be shifted to satisfy the circuit conditions estab- 
lished by the diode. 

Physically, 77 > 0, resulting in an inductive-current decay during diode 
conduction; this decay corresponds to the energy dissipated in the for- 
ward resistance. Since the decay time constant X2 is so very long, the 
load current droops only slightly. 

L 

T2 = zr 
rf 

We must periodically restore the dissipated energy in the inductor's 
magnetic field. On alternate half cycles (negative input), recharge 
takes place, manifesting itself as a small negative excursion of output 
current. The final periodic response is sketched in Fig. 2-21. These 
results are exactly analogous with those obtained in Sec. 2-5 for the volt- 
age clamps. However, whereas recharge in the voltage clamp took place 
during diode conduction, here it occurs when the diode is back-biased. 
The same condition of a large ratio of diode reverses to forward resistance 
is necessary for proper current clamping, and the signal will shift until 
the ratio of the areas equals the ratio of diode resistance. 

Bias currents may be introduced, in parallel with the diode, shifting 
the clamp point wherever desired. As with the voltage clamp, reversing 
the diode reverses the direction of clamping. The nonideal inductance 
has associated coil resistance, which adds to the other circuit resistance 
and decreases both n and t 2 , having the most pronounced effect in short- 
ening the long discharge time constant. 

2-8. Arbitrary Transfer and Volt-Ampere Characteristics. Many 
devices and systems exhibit grossly nonlinear responses which, for 
analysis, we should like to approximate by an equivalent network of 
biased diodes and resistors. These nonlinearities may inadvertently 
arise from the physical properties of the device and will thus represent 
an undesirable characteristic. Alternatively, they may be deliberately 
created and inserted in the signal transmission path in order to perform 
various operations. In both cases generally more than two linear seg- 
ments are necessary for a satisfactory piecewise-linear representation. 
This simply means that additional diodes must be incorporated into the 
network — each break point corresponds to change of state of an ideal 
diode — from conduction to cutoff, or vice versa. One simple example of 
arbitrary transfer and volt-ampere curves and the equivalent diode 
model appear in Fig. 2-22. 



Sec. 2-8] 



DIODE WAVE-SHAPING TECHNIQUES 



49 



If the network contains at most two or three diodes in a relatively 
simple configuration, then, in solving for the break coordinates of any- 
one diode, we can always assume the condition of the others. In Fig. 
2-22, we may assume D\ and D 2 conducting and solve for the break 
values of _D 3 . Any inconsistency arising when checking the answer (e.g., 
a negative current flow through a diode that was assumed conducting) 
means that the problem must be re-solved from the new starting point. 




(a) 



i , tin m a 




8 15 37.5 

i volts »- 

(e) 

Fig. 2-22. An example of a three-diode network together with the transfer and volt- 
ampere characteristics, (a) Three-diode circuit, (b) transfer characteristic; (c) 
input volt-ampere characteristic. 

Three diodes afford eight possibilities as to the circuit condition, but these 
would be halved by the particular diode and biasing arrangement. With 
more than three diodes, in other than a trivial circuit, the trial-and-error 
method becomes unreasonably tedious and we must seek a more organized 
method of solution. 

Example 2-4. In the case where we can readily identify the diode states, as in the 
circuit of Fig. 2-22o, we reduce the complex circuit to a number of simpler models. 
Each region is denned by the resistive network obtained when all back-biased diodes 
are replaced by open circuits and the conducting ones by short circuits. If these 
regions are selected by assuming the diode states, then each model must be checked 
for consistency. In the circuit considered, the region limits are easily specified in 
terms of the output voltage: Di conducts when e„ < 10, D% conducts when e„ > 25, 
and D t is forward-biased when e t < eu- 



50 MODELS AND SHAPING [CHAP. 2 

Region I: 

Di and Z>» back-biased (open circuits) 
Di conducts (short circuit) 

- - *'-*,- ** 10 -8 volts 



" 01 R 1 + R t ' 1K + 4K 

This output remains constant at this value until D s conducts once e^ > 8 volts. 

Region II: 

Di and D% conducting (8 < e„ s < 10) 
Z)j back-biased 

*• - g,\+ ft - + *&+*, ** - °- 286e, ° + 5 - 7 

Region III: Di conducts (10 < e». < 25) 

Di and Ds nonconducting 

e »a "" d — I — 5" e '"> = 0-667ein 
«i -r "4 

Region IV: Ds and D t conducting (25 < e„«) 

Z>i back-biased 

- _ *! II «4 . . + ?'_)'*«- ft - 0.4 eill + 10 



Cot R, || B, + R, Cm T fl» || R,+Rt 

The limits of the output are inserted into the individual equations to find the limits 
expressed in terms of input voltage. These are indicated in Fig. 2-226. Note that 
in each region the transfer characteristic is a straight line with a slope found from the 
incremental model (all battery voltages set to zero). 

A further application of the type of circuit shown in Fig. 2-22a comes 
from recognition that its input volt-ampere characteristic (Fig. 2-22c) 
changes as diodes start and stop conducting. The inverse slope of these 
characteristics is simply the resistance seen looking into the input 
terminals. Therefore diode circuits are useful in developing almost any 
required nonlinear resistance characteristics. Piecewise approximations 
result in a discontinuous rather than continuous incremental input 
resistance, but by taking enough segments we can approximate any 
continuous resistance as closely as necessary. 

A General Approach. Although we are developing more explicit 
methods of analysis and the implicit techniques of synthesis of multiple- 
diode circuits, we shall be forced to restrict the discussion to a single 
reasonably general network. This is essential if the treatment is to be 
kept within bounds; the model used in describing an arbitrary nonlinear 
characteristic is not necessarily unique, and therefore many feasible 
solutions exist. It is neither possible nor practicable to examine all of 
them. 



Sec. 2-8] 



DIODE WAVE-SHAPING TECHNIQUES 



51 



Each branch of the general network of Fig. 2-23a consists of a parallel 
combination of biased diodes and resistors as shown in Fig. 2-23b. The 
over-all response may be determined by noting that: 

1. The number of break points in the transfer or input volt-ampere 
characteristic equals the number of diodes contained in the two branches. 

2. The piecewise-linear response is completely defined by finding the 
coordinates of the break points and the slopes of the two unbounded 
segments. 

3. The maximum possible transfer slope is unity since the network is 
completely dissipative. (We restrict R to positive values.) 



-«12- 



5 



«2 



(a) ~. ■ (b) 

Fig. 2-23. (a) General two-branch network; (6) typical branch configuration. 



In calculating these break points, each branch may be considered 
separately and then their individual responses combined. Under the 
rules developed in Sec. 2-1, the voltage break values of any branch are 
simply the bias voltages appearing in series with the internal diodes. 
Once these voltages are known, the corresponding current values are 
readily found by direct evaluation. It is important to determine which 
diodes are operative in any region, since each one that starts to conduct 
introduces additional shunt resistance and each one that stops conducting 
reduces the number of elements in the branch. 

As an aid to our inconsistent memory, the behavior of each branch 
and the state of each diode should be tabulated as a function of the con- 
trolling terminal voltage. This not only enables us to construct the 
response characteristics in the most organized manner, but in itself 
reflects the many resistive models inherent in the diode network. 

The input volt-ampere characteristic is the totality of the branch 
responses. Under the series arrangement shown, the current must be 
continuous and the voltage drops simply add. The break-point current 
coordinates comprise the set of those found separately for the two 
branches. The corresponding voltage coordinates must be found by 
the direct evaluation and summation of the drops at these known cur- 
rents. For one branch the drop is identically the break value, and for the 



52 



MODELS AND SHAPING 



[Chap. 2 



other it may be determined from the graph or from the equivalent 
resistive network. 

Finally the over-all transfer characteristic is obtained by cross-plotting 
the output branch voltages corresponding to the input break values. In 
each region the slope is identical with that found from the purely resistive 
network holding. Voltage sources only shift the location of the seg- 
ment; they cannot alter its slope. To reiterate, the complete network 
response is determined by the location of the break points and the incre- 
mental slopes of the line segments terminating on them. Only this 
information is necessary for a complete description of the network. 

Example 2-6. The circuit of Fig. 2-24 utilizes series- and parallel-resistance pad- 
ding of Zener diode for its series branch and two individually adjustable diodes for its 




Fig. 2-24. Circuit to illustrate the method of analysis of multiple-diode networks. 





Co> (b) 

Fio. 2-25. Individual branch characteristics for the circuit of Fig. 2-24. (a) Series 
branch; (b) shunt.branch. 



Sec. 2-8] 



DIODE WAVE-SHAPING TECHNIQUES 



53 



shunt branch. The volt-ampere response of each branch may be tabulated with 
respect to decreasing terminal voltage as indicated in Tables 2-2a and 6. Their 
characteristics are also presented graphically in Fig. 2-25o and 6. 



Table 2-2a. Series Branch Response 



Diode states 


ei2 


i, ma 


Incremental resistance* 
T12, kilohms 




en > 



-10 < e 12 < 

-10 

-10 > e» 



-0.5 


4 






Di off, D 2 off 


20 








4 







Table 2-25. Shunt Branch Response 



Diode states 


e 2 


i, ma 


Incremental resistance * 
r s , kilohms 




e 2 > 10 

10 

-10 < d < 10 

-10 

e s < -10 


3 
-1 


2 5 






D a on, Dp off 


5 


D a turns off f 


C.oJ.Djjoff 


10 







* The incremental resistance of each region is simply the parallel combination of all 
resistors inserted by the diodes conducting in that region. 

f The individual branch break points, which are found first, serve as the critical 
points of the table. The voltage values are determined by inspection of the branch, 
and the current by direct evaluation. 



Now that the response of each branch is known, we are in a position to find the 
total input volt-ampere characteristic. This can be done by direct summation of the 
two graphs of Fig. 2-25a and 6 or by considering the data presented in Table 2-2. 
We can construct a new table by interleaving the known break currents in descending 
order (Table 2-3). These represent the totality of break points and separate the 
linear input regions. Furthermore, the incremental resistances previously found are 
in series and may be directly added. The reader should compare the appropriate 
columns of Tables 2-2 and 2-3. 

The only data still unknown are the values of the input voltage coordinates. For 
these we must find the voltage drops across the individual branches at the known 
current values. The derived voltages are given in the parentheses in columns 3 and 
4 of Table 2-3. They may be found by linear interpolation from the tables of the 
individual branch response, from the branch graphics, or directly from the circuit. 

The voltage break value may also be determined by writing and solving the equa- 
tion of a straight line holding within each region. In the region defined by the bot- 
tom line of Table 2-2a, the segment is a straight line of slope ^K passing through the 
point (-0.5 ma, -10 volts). 



54 



models and shaping 
Table 2-3. Total Circuit Response 



[Chap. 2 



Region 


(1) 

Conducting 
diodes and 
diode chang- 
ing state 


(2) 

i, 
ma 


(3) 
en* 


(4) 
e 2 * 


(5) 
ei = ei2 + e 2 


(6) 

Incre- 
mental 
resist- 
ance, 
kilohms 


(7) 

Trans- 
fer 
slope 


I 

II 

III 

IV 
V 


Di, D a , Dp on 
Dp turns offf 
Di, Da on 
Di turns offf 
D a on 

Di turns onf 
D a , Dt on 
D a turns offf 
D 2 on 


3 


0.5 
-1 


(12) 


-10 

(-12) 


10 

(-5) 
(7.5) 
-10 


«i > 22 

22 

-5 > ei > 22 

-5 

-17.5 > ei > -5 

-17.5 

-22 > ei > -17.5 

-22 

d < -22 


6.5 

9 
25 

9 
14 


2.5/6.5 

5/9 

5/25 

5/9 

10/14 



* The derived values are enclosed in parentheses, 
t Circuit break points. 




Fig. 2-26. (a) Input volt-ampere response; (6) transfer characteristics of the circuit of 
Fig. 2-24. 



Sec. 2-9] diode wave-shaping techniques 55 

Thus 

en(i) = -10 volts + (i + 0.5)4 K 

At the known break current of — 1 ma, this yields 

eit(-l) = -12 volts 

Similar equations may be written for the other unknown terms. Figure 2-26a is the 
plot of the complete input volt-ampere characteristics. 

The transfer curve becomes immediately apparent if we recognize that the incre- 
mental transfer slope of each region is 



Aei ri 2 -(- r 2 

where rj and ru are the incremental resistance in the appropriate regions. Table 2-3 
also includes these various slopes. Figure 2-266 shows the cross-plotting of the input 
and output voltage coordinates of the break points leading to the network transfer 
characteristic. The regions are those given in Table 2-3. 

We can draw the following general conclusions from the above example: 

1. Each time a series diode starts conducting, it reduces the net series 
resistance and increases the slope of the transfer characteristics. Each 
time a series diode stops conducting, it reduces this slope. 

2. The conduction of a shunt diode reduces the branch resistance and 
decreases the transfer characteristic slope. As a shunt diode stops 
conducting, both the shunt resistance and the incremental transmission 
increase in value. 

For reference purposes, some commonly used diode-resistor combina- 
tions and their volt-ampere responses are tabulated in Fig. 2-27. The 
first two diagrams illustrate a diode biased to start conducting, and then 
one which stops conducting, as the voltage rises above the threshold 
value. Figure 2-27c represents the response of a Zener diode, and Fig. 
2-27d illustrates the control afforded over the break-point location by 
placing two Zener diodes back to back. This final branch would normally 
be used to insert biased break points into the series arm without having 
recourse to actual bias batteries. 

2-9. Dead Zone and Hysteresis. Two interesting examples of diode 
wave shaping which are used in the analog solution of electrical, chemical, 
and mechanical problems are the generation of the transfer characteristics 
of dead zone and hysteresis. We shall first examine dead zone, which 
appears whenever a threshold value must be exceeded before transmission 
is possible. The steering system of an automobile has built-in dead 
zone for safety; small disturbances of the wheel are not transmitted into a 
turning movement. Some threshold value of steering-wheel rotation 
must be exceeded before the automobile responds. 



56 



MODELS AND SHAPING 



[Chap. 2 



(a) 



<b) 



<e) 



& 



z** 



r 

■5* 



Oi+0z 




Ci+G z 



-B 



r <h 



(d) 



■St Rz 

M - I- -ft 



Si Ez 



fa 



£2 « 



Fig. 2-27. Various diode-resistor combinations and their volt-ampere response. 





Fig. 2-28. Dead-zone circuit and transfer characteristics. 



Sec. 2-9] diode wave-shaping techniques 57 

The freezing-and-melting process of pure materials presents a second 
example of dead zone. Here the addition of heat produces a linear 
temperature rise at temperatures below the critical point. But at the 
melting and boiling points, a finite amount of heat must be added to a 
constant temperature before there will be a change of state (solid to 
liquid, liquid to gas). In the new state, again temperature increases 
linearly with heat flow. 

The simplest diode network which may be used to simulate a fixed- 
width dead zone employs, as the series branch, two Zener diodes connected 
back to back (Fig. 2-28o). Their constant voltage drop in the Zener 
region avoids the addition of any series bias source (a difficult problem 
since this voltage must be isolated from ground). 

We shall consider the response of the nonlinear series branch first. 
On positive voltages conduction eventually becomes possible through 
the forward diode of D x (D v ) and the Zener region of D 2 (D b ). How- 
ever, this will not occur until the branch voltage exceeds E,%. The 
coordinates of the break point are 

ei = E li . i = (2-23) 

At zero current there will be no drop in R 2 and the input break coordi- 
nates are identical with those given in Eq. (2-23). 

By symmetry, we can see that the other break point occurs at 

ei = -E.! i = (2-24) 

Consequently, within the limits 

-E tl <ei< E zi 

only the extremely limited transmission through the high inverse resist- 
ance of a physical diode is possible. Outside these limits the incremental 
transfer slope is determined by the voltage divider composed of R x and 
R2. 

s = sHnr. < 2 - 25) 

In the transfer characteristic of Fig. 2-286 we note that the width 
of the dead zone is a function of the individual Zener diode break voltages 
and that the transmission slope is controlled by the series padding 
resistor. The dead zone may be shifted to any output voltage by suitably 
biasing the output resistor. 

Hysteresis arises in many physical properties of materials, such as the 
magnetization curve of iron and the stress-strain relationship. If the 
forcing function is increased from zero to some final value and then 
removed, the forced system parameter will remain fixed at a value other 



58 



MODELS AND SHAPING 



[Chap. 2 



than zero. But this is simply the object's memory which shows up as 
the magnetic flux still present after the removal of the magnetizing cur- 
rent or as the residual stress in a steel rod after the stretching force is 
released. Of course, limits are imposed by the physical nature of the 
materials. Only so much flux can be supported regardless of the magni- 
tude of the magnetization current; this value is called the saturation 
flux. In a stressed bar, excessive force will cause fracture. 

To return a system (one which may be saturated but not fractured) 
to its original state requires the application of a forcing function in the 
opposite direction, usually of somewhat smaller magnitude than the 
original signal. If the new input is too large, it will simply leave the 
system in an excited state of the opposite magnetic polarity or with 
the strain acting in the opposite direction (Fig. 2-296). 



*i 



9 — VV\ — 



-W- 






D 2f 
-H- 



H °2 



1 



J2t e i2 D 3 y. 

+ 1 
E m -=- 



¥ 



C±Z e 2 




(a) (b) 

Fig. 2-29. Hysteresis circuit and characteristics. 



A smaller amplitude excitation will generate smaller area loops with 
regard to both the displacement from the origin and the maximum output 
amplitude. These are extremely difficult to simulate, and therefore we 
shall concentrate our attention on the circuit of Fig. 2-29a, which gives 
the response shown in Fig. 2-296. The two diodes D 3 and Z> 4 limit the 
maximum output excursion to ±E m and establish the saturation values 
so necessary in many hysteresis systems. (Fracture may be simulated 
by fuses.) 

Operation is similar to that of the dead-zone circuit. On a rising 
input, the forward diode D lf and the Zener diode D iZ eventually conduct, 
providing an output across C. Capacitor voltage then follows the input 
to a maximum value determined by the clipping diode D 3 . As the input 
falls, the charge on the capacitor (the memory of previous maximum 
excitation) back-biases the formerly conducting diodes and, since no 
discharge path exists, the output remains at the maximum value pre- 
viously reached. Eventually the falling input brings D xz and D if into 
conduction, establishing a discharge path, and again the output follows 
the input. The transfer characteristic now traces the left-hand portion 



Sec. 2-10] diode wave-shaping techniques 59 

of the curve. If the input continues falling and then rises, the cycle 
repeats, with the roles of the series diodes interchanged. 

There is no necessity to drive the output into saturation. For genera- 
tion of hysteresis or backlash curves, smaller inputs generate the minor 
loops indicated by the dashed lines of Fig. 2-29b. 

The slowest input signal for proper response is determined by the 
leakage of the capacitor charge through the diode inverse resistance, and 
the fastest signal by the charging time constant 

We choose C for reasonable response to the expected time range of the 
input signals. 

2-10. Summary. Many nonlinear systems whose externally measured 
characteristics are available may be treated from the viewpoint of their 
piecewise-linear representation. The nonlinear system, difficult to treat 
in all its complexity, is reduced to a sequence of linear problems, each 
with its own response and limits of operation. Simple mathematical 
tools already at our disposal from studies of linear systems are sufficient 
to reach acceptable engineering answers in those cases amenable to this 
treatment. We can always handle circuits involving only dissipative 
elements, no matter how nonlinear their characteristics may be. When 
energy-storage elements are also contained in the nonlinear system, the 
problem becomes much more difficult. 

If in each region the boundary conditions are independent (able to be 
calculated only from the model and the boundaries of the region), 
we can solve the system in a straightforward manner. If the constraints 
involve past history, as in the hysteresis circuit, then when only a single 
energy-storage element is present, we are able to complete the analysis. 
And if multiple modes of energy storage existed, the circuit would not 
have yielded so readily to simple methods. Except in very special cases, 
the problem faced is extremely difficult and beyond the scope of this text. 

The first step in any analysis is the construction of the appropriate 
models together with the clear delineation of their boundaries. Gener- 
ally, boundaries are determined by the energy-storage elements and points 
of conduction. Secondly, conditions of continuity across the region's 
boundaries must be considered. These arise primarily from the original 
characteristics and the type of energy-storage elements involved, with, 
however, the energy-storage elements often determining the diode con- 
ditions. And finally we are ready to proceed to the solution. 

The same techniques are used in solving many chemical, thermal, and 
mechanical systems, where, for example, an ideal mechanical "diode" 
might be postulated rather than an electrical one. A velocity-operated 
clutch which engages or disengages at present shaft speeds is one mechan- 



60 MODELS AND SHAPING [CHAP. 2 

ical example. Its slip on engagement is analogous with the diode for- 
ward resistance, while its imperfect disengagement represents reverse 
resistance. In fluid flow, on-off pressure valves behave similarly, with 
their pressure loss and leakage modifying the ideal valve in much the 
same way as resistance modified the ideal diode. A metal plate, polished 
on one side and black on the other, allows radiant heat to flow more read- 
ily in one direction than in the other; it may be used as our thermal diode 
under the applicable conditions. 

In many cases where it is difficult to formulate proper models, making 
the grossest approximations leads to a circuit which can be solved. The 
answer, admittedly incorrect, often gives some insight. This enables 
us to refine the analysis later, eventually coming within an acceptable 
engineering solution. 

PROBLEMS 

2-1. (a) By direct graphical construction evaluate the input volt-ampere charac- 
teristics of Fig. 2-30. Specify the individual slopes and the coordinates of the single 
break point. 

(6) Convert the circuit holding for each diode state into both Thevenin and Nor- 
ton equivalent circuits and find the break point from the two defining equations. 




>2K 



X lOmafrj 

SI J 



Fio. 2-30 

2-2. Show how the transfer characteristics of Fig. 2-8 may be derived by a graphical 
construction. (The volt-ampere characteristics of the input and of the shunt diode 
branch must first be evaluated.) 

2-3. A diode clipper such as the one shown in Fig. 2-7 is used in a system where the 
input may vary from zero to a value that will never exceed four times the clipping 
level. What series resistor should be chosen, expressed in terms of ry and r„ so that 
the maximum error voltage on both sides of the clipping value will be equal? 

2-4. A triangular wave such as the one shown in Fig. 2-15 (40 volts peak to peak) 
is the input to the series clipper of Fig. 2-10. The bias battery is +5 volts, and the 
diode parameters are r r = 100 K and r/ = 40 ohms. Sketch the transfer character- 
istics and draw the input and output waveshapes to scale when B\ equals (a) 25 K, 
(6) 4 K, and (c) 1 K. In all cases calculate the peak values of the output signal. 

2-6. Repeat Prob. 2-46 for the circuit of Fig. 2-9 where the +5-volt bias is derived 
from a tap on a 10,000-ohm bleeder connected across a 50-volt power supply. 

2-6. We wish to clip the positive peaks of a signal at +10 volts by using a shunt 
clipper similar to that shown in Fig. 2-7. The available diode has a forward resistance 
of 100 ohms and an inverse resistance of 0.5 megohm, and in addition it has a Zener 
break point at —25 volts. 



DIODE WAVE-SHAPING TECHNIQUES 



61 



(a) Calculate the optimum series resistance and draw the circuit. 
(6) Sketch the transfer characteristics, indicating the values of slopes and the 
coordinates of all break points. 

(c) Draw the output waveshape when the driving voltage is a 100-volt peak-to- 
peak sinusoidal signal. 

(d) Under the conditions of part c, what angular percentage of the sine wave would 
be transmitted relatively unaffected by the clipping circuit? 

2-7. The single pulse shown excites the circuit of Fig. 2-31 at t = 0. If the initial 
charge on C is —20 volts (polarity shown), evaluate the complete output response. 
Specify all voltage and time-constant values. Make all reasonable approximations. 

100 K 



+50 v 



500 
/(sec 




Fio. 2-31 



2-8. A 30-volt peak-to-peak 100-Msec-period square wave is clamped negatively at 
+5 volts. The diode has a forward resistance of 50 ohms and an inverse resistance 
of 100 K. We want to use the smallest possible capacitor so that this clamp will have 
a fast response to changes in the input-signal level, but we are absolutely restricted in 
that the output must never exceed the clamping level by more than 2 per cent of the 
peak-to-peak input signal. 

(a) What is the smallest-size capacitor which we may use? Round off your answer 
to the larger even value, for example, 0.00228 == 0.0025. Explain why you would 
use a larger value. 

(6) How much energy must be restored in the capacitor over each cycle? (Make 
any reasonable approximations to simplify your calculations.) 

(c) If the peak-to-peak signal is suddenly increased to 50 volts, how many periods 
of the input signal will the circuit take to recover its clamping action? 

(<Z) Repeat part c if the signal is decreased to 10 volts. (Hint: Consider the super- 
position of the square-wave input and the initial voltage across C. Find the time at 
which the diode again conducts.) 

2-9. The signal of Fig. 2-32 is the input to a clamp having r r = 1 megohm, 
r, = 50 ohms, and C = 0.005 4. 

(a) Sketch the output if this signal is clamped positively at zero and indicate all 
important voltage values and time constants. 

(6) Sketch the output if this signal is clamped negatively at zero. 

(c) Under the conditions of part 6, how long after the termination of the large pulse 
before the clamp capacitor is again charged? (See hint given in Prob. 2-8d.) 




6 8 10 12 14 16 18 20 2 



Fig. 2-32 




t, msec 



62 



MODELS AND SHAPING 



[Chap. 2 



2-10. Solve the circuit of Fig. 2-15 for Ei, E' 2 , Ti, and Tt where r, = 10 ohms, 
r r = 200 K, R, = 1 K, where the triangular wave period is 10 msec. Assume that C 
is very large. 

2-11. Consider the response of the circuit of Fig. 2-33 to the signal given. The 
input has been at 200 volts for a very long time before the negative pulse appeared. 
Draw the output waveshape, indicating all time constants and voltage values. 

0.005 ^f 



«1n, 


I 






ZOOv 










~l 




100 

o 









1 
1 
I 






2 t, msec 








Fig. 2-33 



im: 



+300 



1 K< 



2-12. Draw a circuit that will clip the positive peak of a 100-ma peak-to-peak tri- 
angular driving current at the 20-ma point. Use a diode having a forward resistance 
of 50 ohms and r, = 100 K and specify the optimum shunt resistance. Sketch the 
output current, giving all important waveshapes. Explain how an other-than-zero 
load resistance would affect the clipping action. 

2-13. The current waveshape of Fig. 2-34 is to be clamped negatively at —5 ma. 
The diode available has a forward resistance of 500 ohms and an inverse resistance of 
1 megohm. The coil approximates an ideal inductance in that its resistance is only 
10 ohms and its inductance is 50 mh. 

(a) Draw the circuit used, showing where the bias current would be injected. 

(6) Sketch the output current, indicating all important values and time constants. 

(c) To what should the spacing between the pulses be changed so that the output 
overshoot will just equal 1 per cent of the peak-to-peak current? 

(d) Repeat part 6 if a 500-mh coil is used in place of the one specified above. 



50 
ma 



t, 
Msec 



5 



60 55 
Fig. 2-34 



100 105 



2-14. What effect would the coil resistance Bl of a nonideal inductance have on the 
current clamping action? Consider clamping a square wave of current at zero to 
simplify your analysis. 

2-15. Prove that the output waveshape of a current clamp will shift until the ratio 
of areas is equal to the ratio of diode resistances. 

2-16. A 20-volt peak-to-peak 1-msec square wave is the input to the clamp of 
Fig. 2-12. The diode has a forward resistance of 20 ohms and an inverse resistance 
of 100 K; C = 0.01 rf. 

(a) Sketch the output waveshape for the first two cycles if the square wave is 
initially applied when it is zero, going positive. 

(6) Repeat part o if a 0.001-/if capacitor is shunting the diode. Make all reason- 
able approximations. 



DIODE WAVE-SHAPING TECHNIQUES 



63 



2-17. Given a d-c vacuum-tube voltmeter with a 10-megohm input impedance, our 
problem is to construct an a-c voltmeter. The d-c unit reads correctly in conjunction 
with a 1-megohm probe. Two different a-c probes are under consideration (Fig. 
2-3Sa and b). In each case they will be connected directly to the voltmeter, i.e., not 
through the d-c probe. 



0.05/»f 

-K- 



Input 



5.5 M 
-VW- 



To 
VTVM 




Input 



VTVM 



(a) 



(b) 



Fig. 2-35 



Calculate the VTVM reading for each probe for the various inputs listed below. 
In each case relate the reading to some parameter of the input signal (rms, peak, or 
average). 

(a) 10-volt-peak 1,000-cps sine wave. 

(6) 10-volt-peak 1,000-cps square wave. 

(c) 20-volt peak-to-peak 1,000-cps triangular wave having + 10-volt average value. 

(d) Signal same as in part c except that it has a — 10-volt average value. 

(e) 10-volt 10-/isec-wide positive pulses with a period of 100 /isec. 
(J) Repeat part e for negative pulses. 

2-18. Plot the transfer and input volt-ampere characteristics of the circuit given 
in Fig. 2-36. Sketch the output voltage if em = 100 sin at, giving all important values 
(assume ideal diodes). 

-H- 

50 K 



50 K 
-WW- 




:iook 



25 K' 



!5v 



Fig. 2-36 



Fig. 2-37 



2-10. Draw the input volt-ampere and the transfer characteristics of the circuit of 
Fig. 2-37 and sketch the output when a 50-volt peak-to-peak triangular wave is 
impressed at the input. 

2-20. (a) Set up the equations representing each region of operation given in 
Table 2-3 (Sec. 2-8) and verify the values of all break points. 

(6) Show the purely resistive model (TheVenin or Norton equivalent) implicit 
within each region and specify the various component values. By direct solution of 
the intersection of the models found, evaluate the break points bounding these regions. 

2-21. Design a circuit which will have the transfer characteristics given in Fig. 2-38. 
Specify all batteries and resistors, taking the smallest resistor as 10 K. 



64 



MODELS AND SHAPING 



[Chap. 2 




Fig. 2-38 

2-22. In the circuit of Fig. 2-39, the input voltage is a symmetrical triangular wave 
having a period of 16 sec, a peak-to-peak amplitude of 80 volts, and an average value 
of zero. 

(a) Plot the output voltage as a function of time and label all break points with 
their time and voltage values. 

(6) The output signal is to be a rough approximation of a sine wave, with only the 
break points agreeing exactly with the sinusoidal signal. What is the peak amplitude 
of the sine wave we are approximating, and to what angles do the break points 
correspond? 



100 K 
<?-vW- 



eti 



50 K 
-►HWV-O+20V 



50 K 
-H^VWo-20v 




._ 200 K tj ,200K 

Fiq. 2-39 

2-23. One method of multiplying two voltages together is to convert each voltage 
signal into a current proportional to its logarithm (to the base 10) and then to add the 
two currents in an arrangement similar to the summer of Chap. 1. If the output 
voltage is kept small compared with the input, then we might employ a circuit whose 
volt-ampere characteristics are logarithmic to reconvert log A + log B to the product 
AB. This basic circuit is shown in block-diagram form in Fig. 2-40. 



ij«loge J 



eo^me^ 



i 2 a\oge 2 



I 



Fig. 2-40 

(a) As the first step, design a diode circuit which will approximate the logarithmic 
volt-ampere response for positive input signals. Except for a scale factor, the input 
current should exactly agree with log e ln at the following integer voltages: 10, 20, 40, 
70, and 100 volts. Use 100 K as your smallest resistance value and specify all resistors 
and bias sources. 



DIODE WAVE-SHAPING TECHNIQUES 



65 



(6) If we use tables to reconvert the sum of the logs found from the two circuits of 
part a, what error is introduced when multiplying the following numbers: 20 X 65, 
20 X 50, 12 X 30? 

(c) If we use a circuit identical with that developed in part a (except that the 
smallest resistor is now 1,000 ohms) for reconversion of the sum of the logs to the 
product of the input voltages, then 

e„ ~ meiet 

What is the circuit constant ml To what values would the errors of the products of 
part b increase as a result of this additional circuit? 

2-24. (a) Discuss the effects on the transfer characteristic of the circuit of Fig. 2-28 
when we return the bottom of Us to a variable bias instead of directly to ground. 

(6) A 100-volt peak-to-peak sinusoidal signal is applied at the input of this circuit. 
Sketch the input and output to scale and determine the transmission angle if the fol- 
lowing circuit components are used : 



l-E,. 



\E.,\ = 20 volts fii = 10 K R t = 100 K 



2-25. Prove that the circuit of Fig. 2-41 also simulates a dead zone. Calculate the 
transfer characteristic and specify all break points and slopes. What would happen 
to the transfer characteristics if diode Z)i is transferred from point A to point B? 



<f+ 50 v 
►20 K 



A] 
1K< 






1K< 



1 



2 

-w- 



:im 



;20K 
50 v + 



Fig. 2-41 



I 



2-26. An 80-volt peak-to-peak symmetrical triangular wave is impressed at the 
input of the hysteresis circuit of Fig. 2-29. The dead zone is symmetrically located 
about the origin and has a width of + 10 volts. Sketch the input and output to scale 
if the circuit saturates at ±30 volts. Ri is very small, and C is very large. 

2-27. The circuit of Fig. 2-29 may also be employed as a memory, as may any 
device exhibiting hysteresis. If the saturation region crosses the vertical axis, then 
once the input drives the output into saturation, the output will remain there even 
after the removal of the input pulse. Only by injecting a pulse of the opposite polarity 
can the circuit be reset to the other saturation value. 

(o) Design a circuit which will saturate at ± 10 volts and which will cross the «i = 
line at this value. The only two available voltages are ±45 volts. Specify all 
resistors and draw the transfer characteristic. 

(6) Explain the disadvantages of setting the crossover point on the sloping side of 
the transfer characteristic instead of in the center or near the center of the saturation 
region. 

(c) Sketch the output waveshape if the input pulses are those shown in Fig. 2-42. 



66 



MODELS AND SHAPING 



[Chap. 2 



+30 v 

t 
e 




































20 25 




40 45 50 55 






5 10 15 




30 35 








60 65 t, msec 


-30 































Fig. 2-42 

2-28. The temperature-vs.-heat-content curve of many pure Substances exhibits 
one or more dead zones. These occur where heat of fusion or heat of vaporization 
must be added to cause a change of state: solid to liquid or liquid to gaseous. For 
example, the temperature of ice increases linearly with heat content from —20 to 0°C 
and then remains there until sufficient heat has been added to melt all the ice. Tem- 
perature again increases with the addition of heat, but at a new rate to the boiling 
point. 

(a) We are interested in developing a circuit to represent this action where the input 
voltage would correspond to the heat content (each volt to 2 cal) and with the output 
voltage corresponding to the temperature (5°C/volt). Choose water at 0°C as the 
origin of the axis and calculate the slopes, intercepts, and voltages needed to represent 
a temperature range from —20 to 90°C. Design the diode circuit which will give the 
required transfer characteristic. Try to use the minimum number of components. 

(b) Repeat part o for a temperature range from —20 to 110°C. Choose a new 
calorie-temperature conversion ratio. 



BIBLIOGRAPHY 



Angelo, E. J., Jr.: "Electronic Circuits," McGraw-Hill Book Company, Inc., New 
York, 1958. 

Millman, J., and H. Taub: "Pulse and Digital Circuits," McGraw-Hill Book Com- 
pany, Inc., New York, 1956. 

Stern, T. E. : Piecewise-linear Network Theory, MIT Research Lab. Electronics Tech. 
Rept. 315, June 15, 1956. 

Zimmermann, H. J., and S. J. Mason: "Electronic Circuit Theory," John Wiley & 
Sons, Inc., New York, 1959. 



CHAPTER 3 

DIODE GATES 



In contrast to Chap. 2, which treated diode networks excited at a 
single terminal, this chapter will discuss circuits designed for operation 
with multiple inputs. Where previously the complete circuit response 
was characterized by the transmission and volt-ampere curves, this 
simple representation now becomes quite inadequate. The multiplicity 
of inputs interact, and consequently the state of any particular diode, at 
any particular time, cannot be defined in terms of one input but rather 
depends on the relative effects of the several sources. It is completely, 
possible for a signal applied at one set of terminals to control the trans- 
mission path between another terminal and the output. Since it is 
impossible to solve this problem in completely general terms, we shall 
at least try to answer the following three questions in the course of the 
analysis of specific circuits: 

1. Does a transmission path exist from any one terminal to the output? 

2. What conditions must be satisfied elsewhere in the network to 
establish (or to interrupt) this path? 

3. How does the circuit behave in the two regions separated by the 
transmission threshold? 

3-1. Application. Gates are circuits which make use of the bistate 
diode properties for switching purposes. Defined, not in the narrow 
sense of turning something on and off, even though this is included, but as 
the establishment of information transmission paths upon the application 
of a proper stimulus, diode gates find wide application in digital com- 
puters, control and measuring instruments, and stimulation systems. 
In computers the gate output amplitude is usually of little or no impor- 
tance provided that it at least exceeds a preset threshold value. We 
consider the output's presence as representing a "yes," or 1, answer, with 
its absence as a "no," or 0, answer. Almost any desired information 
for computing or control can be conveyed and operated upon by passing 
a coded time sequence of pulses through various combinations of gates, 
each pulse carrying one item of information. 

In special-purpose instruments, an external control signal, or proper 
sequence of signals, opens and closes the gate, thus allowing or preventing 

67 



68 



MODELS AND SHAPING 



[Chap. 3 



transmission of the information signal. Here the amplitude response 
becomes important because this information may later be used for con- 
trol purposes. 

Diode gates are classified, according to their performance, into three 
groups : the or gate, which has an output when any one or all inputs are 
present; the and gate, which allows an output only when all inputs are 
applied; the controlled gate, where a controlling signal turns on the gate, 
allowing transmission upon satisfaction of its other requirements. 

Gate inputs are often short pulses. Consequently, to avoid introduc- 
ing reading errors, the gate should open fast to transmit the individual 
pulse and close fast upon its removal to prevent false information from 
appearing at the output. Any delay in opening and closing creates an 
ambiguity in performance; therefore our investigation of the particular 
circuits must be concerned with both their amplitude and time response. 




t 

«1 










t 

«2 






1 1 
1 1 
! > ' 

1 1 ' 
1 1 1 


t 


1 


II II 

! ! ! ! 




t 


1 
*3 


1 
1 

1 

1 






1 


! 

1 — 1 




t 


1 






t 



(a) (b) 

Fig. 3-1. (a) Three-diode or gate; (6) input-output relationships. 

3-2. oe Gate. The circuit of a simple on gate designed for operation 
with positive input pulses appears in Fig. 3-lffl. For negative inputs 
all diodes are reversed. The time relationship of the various inputs 
to the output is illustrated in Fig. 3-16, ideal gate operation assumed. 

For the initial discussion assume that the circuit is ideal; i.e., all source 
impedances (r,i, r„2, and r, 3 ) are zero and all diodes switch between open 
and short circuits. Suppose that at some instant ei > and ei = e 3 = 0, 
the single diode D\ associated with e x conducts, establishing the trans- 
mission path between input and output. The output, now equal to e\, 
is larger than d and e s , back-biasing their associated diodes. Only the 
single source ei supplies power; it sees the load R. All other signal 
sources work into the open circuits of nonconducting diodes. 

When all inputs are greater than zero, the output is equal to the largest. 
To prove this, apply the signals e x > e 2 > e 3 > and assume that 



Sec. 3-2] diode gates 69 

initially D 3 conducts. The resultant output e 3 is less than ei and e 2 , 
forward-biasing their diodes. Thus there appears to be a transmission 
path for each of the inputs. However, this argument is fallacious: once 
Di conducts, the output reaches ei, the highest voltage of all applied 
signals, back-biasing the diodes D 2 and D 3 . Since we can only maintain 
the single transmission path through Di, the output of the or gate auto- 
matically becomes equal to the largest input signal at each instant of time. 

The or gate may be considered analogous to a group of normally 
open parallel relays or switches connected from various signal sources 
to the common output. Closing any one provides an output regardless 
of the state of any of the others. A mechanical system of several power 
sources coupled through clutches to the same drive shaft behaves sim- 
ilarly. The engagement of any single clutch furnishes rotational power, 
and if ratchet or slip clutches are used, only the one connected to the fast- 
est-rotating power source transmits power. All others slip (back-biased). 

Often a number of instrument or telemeter outputs whose source 
impedance may be too large to be ignored are the signals driving the gate. 
Before we can begin calculating the corrected amplitude response, we 
would like to know each diode's state so that it can be replaced by 
either an open or a short circuit. If only one input is applied, this 
problem is trivial. But suppose that all inputs are simultaneously 
excited by voltages of the same order of magnitude; how do we now find 
each diode's state? We might, as a first attempt, guess that they all 
conduct. Then, by superposition from the model found by replacing all 
diodes by short circuits (r, 3> r t ) , the output is 

„ „ «||r.,||r., m R 11 r.i || r. z , R \\ r.i H r„ 



r.x + R || r. 2 || r„ B1 ^ r„ + R || r.x || r., "' ' r„ + B || r fl || r„ 

(3-1) 

The next step is to compare e„ found above with each input (ei, e 2 , e 3 ). 
If it is larger than one or more, then the diodes connecting the smaller- 
amplitude generators are back-biased and those particular inputs cannot 
contribute to the output. For example, when e„ < ei, e„ < e 2 but e„ > e 3 , 
we must modify Eq. (3-1) by opening the transmission path from e% 
to the output. The output now becomes 

J _ B||r rt ei+ g|K» 62 (3.2) 



~ r.i + R || r.^ 1 ^ r.. + R \\ r., 

(Jnder the circuit conditions leading to Eq. (3-2), signal sources ei 
and e 2 are loaded by both R and each other; e% isolated by its back-biased 
diode need not be considered. The signal power requirements are deter- 
mined by the current each source must supply. This depends upon the 
difference between the open-circuit terminal voltage and the output 



70 



MODELS AND SHAPING 



[Chap. 3 



voltage, and thus upon the contributions from all other signals, 
example, the current flow from e x is 

Ci — e 



For 



»i = 



r.i 



If the or gate contains more than the three sources shown, the addi- 
tional ones are treated similarly, contributing to the output only when 
their particular diode conducts. 

Transient Response. Our investigation of the transient response of the 
or gate will initially concern itself, for simplicity, with the two-diode 
gate shown in Fig. 3-2. High source impedance degenerates the pulse 
waveshape by increasing the circuit time constants; if we are interested 
in fast response, the gate must be driven from such low-impedance sources 
as cathode followers or transformers. 

|f 




-H- 




->r- 



— Wv- 

-H — 

r r 
-WV 

-VvV- 






Fig. 3-2. Two-diode or gate — piecewise-linear circuit. 

In the above circuit the diode shunt capacity, only about 1 or 2 /x/if , 
is obviously much less than the total stray output capacity (10 to 50 wi). 
By neglecting the small term, we avoid the necessity of calculating the 
response of a circuit containing two independent energy-storage elements, 
the series combination of r.j and C 2 in parallel with R and C . Further- 
more, a gate should not attenuate the input pulse excessively while 
transmitting it. This requires that (r, || R) » (r f + r.i). Under these 
conditions the output's peak value will be almost the same as that of the 
single input, rising toward this peak with a time constant t\, which is 
found from inspection of the simplified model. 

ri = [(r.i + r,) || R || r r ]C. S (r.i + r f )C, (3-3) 

After the input pulse disappears, the output decays toward zero with 
the much longer time constant 

(3-4) 



- = («U r i)' 



Sec. 3-2] 



DIODE GATES 



71 



Figure 3-3 shows the applied input and the resultant output pulse. 
We might observe that for an extremely narrow input pulse, e might 
never reach the final value and might not even pass the arbitrarily 
designated threshold before the input drops to zero. In addition, the 
long decay time constant smears the trailing edge, causing the output 
to persist for some time after the input's removal, a false indication that 
the input is still present. We see that the gate's resolution is severely 
limited by its time constants; it cannot distinguish extremely narrow 
pulses. 

t 







t 

e 


r 


\ 








Threshold 








*- 


I 




s. 



l-«-At^ t ' k-At-H t 

Fig. 3-3. OR-gate input and resultant output pulse. 



I — VW-A/W 




Fig. 3-4. n-diode-gate equivalent circuit — single-diode excitation. 

Computer or gates, receiving information pulses from many sources, 
usually contain appreciably more than the two diodes shown in Fig. 3-2. 
We suspect, even before we begin calculating the response, that the 
additional ones will affect the pulse waveshape through the increased 
resistive and capacitive loading introduced. In any or gate, when only 
one diode is excited, all the others are back-biased and in parallel. 
This is indicated in the equivalent circuit of an n-diode or gate (Fig. 3-4). 
Furthermore, it might reasonably be expected that, in any one system, 
the inputs would be supplied from similar sources to nearly identical 
diodes. Since all the back-biased diode branches are identical, the same 
voltage must be developed across each branch element, thus allowing us 
to connect equivalent elements together (the dashed line of Fig. 3-4). 

By doing this we parallel all source impedance (r,i, r.j, . . . , r,„); 
and as n increases, this combination decreases proportionally, approach- 
ing zero as a lower limit. Even though it is difficult to justify the 



72 MODELS AND SHAPING [CHAP. 3 

omission of a single source impedance, it is relatively easy to justify the 
neglect of the very-low-resistance parallel combination. The reduoed 
circuit places all capacity in parallel, becoming a single-energy-storage- 
element system. Furthermore, R || [r r /(n — 1)] » (r.i + r f ), prevent- 
ing excessive pulse attenuation by the gate. As a consequence, 

n £g (r.i + r f )[C t + C d {n - 1)] (3-5) 

where C d is the shunt capacity of the individual diode. The increase 
in time constant, completely due to the diode capacity loading, slows 
the initial rate of pulse rise. Therefore, if narrow pulses are to be trans- 
mitted, we must limit the number of diodes in any one gate. 

After the termination of the pulse, the energy stored in the output 
capacity back-biases all diodes and the output decays with the slow time 
constant 

r 2 £ R || J (Co + nCi) (3-6) 

For n sufficiently large, the circuit resistance during decay is primarily 
determined by the reverse resistance of the parallel diode. It follows 
that Eq. (3-6) may be rewritten 

nSC.J +C d r r (3-7) 

The first term of Eq. (3-7) is obviously much less than the two-diode 
decay time constant [Eq. (3-4)], and it further decreases with an increas- 
ing number of inputs. The second term is the very small recovery time 
constant of the individual diode. We come to the surprising conclusion 
that the additional diodes may even improve recovery through reduction 
of the circuit time constant. This is the direct consequence of the circuit 
resistance decreasing faster than the capacity increases. However, the 
greater loading of signal source has the adverse effect of increasing both 
the attenuation of the pulse and the source power requirements. 

3-3. and Gate. The and gate determines coincidence, presenting an 
output when, and only when, all inputs are simultaneously excited. Its 
operation may be compared to a parallel combination of normally closed 
switches shorting the output to ground; only after opening all switches 
will the output be other than zero. In hydraulics, the analogous circuit 
is a series of check valves, all of which must be opened before any fluid 
flows from the source to the sink. 

A simple diode and gate operating on positive pulses appears in Fig. 
3-5a, with the input-output time relationships shown in 3-56. Since 
transmission of negative pulses is possible without coincidence of inputs, 
we include at the output a diode D whose function is to prevent negative 



Sec. 3-3] 



DIODE GATES 



73 

all diodes and bias 



excursions. For operation with negative inputs, 
voltages are reversed. 

When ei and/or e 2 are zero, at least one diode conducts, shorting the 
output to ground. (Ideal diodes and zero source impedance, as shown in 
Fig. 3-5o, are assumed for this initial discussion.) However, if signals 



44- 



D 2 
-M- 







! 

e 2 

t 

e 






t 








l_ 


1 


t 

*■ 

t 



(a) (» 

Fig. 3-5. (o) Ideal and gate; (6) AND-gate input-output relationships. 

are applied so that Eu> > ei > e 2 > 0, only diode D 2 conducts, and it 
connects the output terminal directly to the source e 2 . Diode D\ is 
back-biased, removing ei from the circuit. The output, consequently, 
will always equal the smaller of the ivputs. A second possibility arises 
where both input signals are greater than Em,, back-biasing both diodes. 
In this case the output rises to En, its 
maximum possible value. 

Transient Response. Having es- 
tablished the basic behavior of the 
and gate, we are ready to proceed 
to the quantitative analysis of the 
possible modes of operation defined 
above, using, for this, the complete 
model of Fig. 3-6. 

Case 1: E» > ei > e 2 > 




Fig. 3-6. Model of an and gate showing 
parasitic capacity and resistance. 



Here the output rises to approxi- 
mately e 2 from an initial voltage 
slightly above zero (due to the drop 
across r, + r/). Exact initial and 
final values are easily found by drawing and solving the resistive models 
holding. Before the pulses are applied, both diodes are conducting, but 
after reaching steady state, only D 2 conducts. Even a superficial exam- 
ination convinces us that the condition necessary to minimize the small 



74 MODELS AND SHAPING [CHAP. 3 

initial value and simultaneously to maximize the final value is 

r. 2 + r, « R « j (3-8) 

Since the stray capacity prevents any instantaneous change at the 
output, upon injecting the input pulses all diodes are immediately driven 
off. The output rises toward En, with the time constant 



n = ( R I! ^j (C. + ZC d ) (3-9) 

However, it will never reach Ebb) before it can do so the output will 
equal the smaller of the two inputs, e 2 . D 2 is again forward-biased, 
and it limits the rise. At this point the charging suddenly ceases. The 
time required for the complete rise is given by 

h = ti In Ebb (3-10) 

atb — e 2 

When the smallest pulse is 30 per cent less than En,, this time is only 
1.2ti. Limiting e 2 to an even smaller percentage of Eu, will further reduce 
the circuit response time : with a pulse of 0.5En, the rise time is appreci- 
ably less than one time constant; it is only 0.69n. 

As soon as a single input returns to zero, the connecting diode again 
conducts and, using the inequality of Eq. (3-8), the output decays with 
the time constant 

r 2 = (r„ + r,){C, + 20,) (3-11) 

Obviously, the decay through the on diode is much faster than the initial 
rise. 

Case 2: ei > e 2 > Em, 

Under these circumstances the output will always be less than the 
amplitude of the applied excitation. The coupling diodes are initially 
driven off and remain in this state until the input pulses disappear. 
There will be no shortening of the output rise, which, within approxi- 
mately 4ri, reaches Em,. Even if the rise time is defined as in Eq. (1-32) 
(the time for the output to rise from 10 to 90 per cent of Ebb), it will still 
take h = 2.2ti sec to reach an acceptable final value. 

The decay in this case is identical with that found for case 1. 

A graphical cbmparison of these cases, where the value of Ebb is adjusted 
so that both outputs are of the same amplitude, appears in Fig. 3-7. 
We conclude, from this sketch and from the relative rise times calculated 
above, that case 1 is the preferred class of operation, especially if we can set 
Ebb well above the pulse amplitude. It also offers other advantages 



Sec. 3-4] 



DIODE GATES 



75 



in that the output pulse is not clipped during transmission and that it is 
available at low impedance (through the conducting diode) for direct 
application at the input of the next gate. 

Additional input diodes change the gate response primarily by increas- 
ing the stray capacity and reducing the effective reverse resistance 
during the initial pulse rise. Arguments similar to those used with the 
or gate lead to the same general conclusion: when extremely narrow 
pulses are the gating signals, the number of diodes in the gate must be 
limited; otherwise the gate will distort the pulse beyond all. recognition. 



E t 



Co 


/ 
/ 

1 


1 



e 2 1—; 



t, 
Case 1 
Fig. 3-7. Comparison of the output pulses under the two modes of AND-gate operation. 







t 



C 2 * D 2 



Cc ^ At 



", innn 



in 






— »» 
t 


n 


n 


n 


t 


n 


n 




t 



Fig. 3-8. Controlled gate, and-or, and the applied inputs and resultant output. 

3-4. Controlled Gates. Almost any method of either shifting the 
d-c operating point of the gate or applying the control voltage in such a 
manner that it prevents transmission converts an ordinary and or or 
gate into a controlled one. An extremely simple change results in 
the circuit of Fig. 3-8, where the pulses are a-c-coupled rather than 
d-c-coupled as previously. Changing the input circuits back to the 
direct coupling used in Sees. 3-2 and 3-3 will not change the basic opera- 
tion; it only changes the required driving signal. 



76 MODELS AND SHAPING [CHAP. 3 

The two input diodes constitute a simple or gate transmitting to 
point b any positive pulses applied at e x and/or e 2 . However, if D is 
kept conducting, the output pulse will have to be developed across its 
very low forward resistance. Consequently, the output remains close 
to zero regardless of the signals ei and e 2 . The additional control diode 
D c and its very negative bias voltage E ce serve to maintain this condition 
by establishing a large forward-current flow. Diode D also performs 
the second function of preventing negative excursions at the output. 
A large positive control pulse applied at e c back-biases the previously 
conducting control diode and keeps it back-biased until C c can recharge 
through R e . This pulse effectively disconnects the bias voltage from the 
output, allowing transmission of any positive pulses now applied at 
e\ and/ or e 2 . We see, therefore, that the condition for opening this gate 
is the simultaneous excitation at e c and e\ or e 2 . 

Unless the input-circuit time constants are very long in comparison 
with the pulse duration, they will distort the applied signals, leading to a 
correspondingly poor output waveshape. 

Returning the control diode of Fig. 3-8 to a positive bias voltage En, 
rather than the negative one shown, changes this gate into a not-or 

circuit. The positive voltage nor- 
\ bb mally back-biases D c so that it 

plays no role when pulses are 
applied at ei and/or e 2 . But if 
simultaneously we apply a larger 
i t * h? . negative pulse at e c , it establishes 
a conduction path through D„ and 
, D c , maintaining the output at zero, 
independent of ei and e 2 . This 
negative pulse is sometimes referred 
Fig. 3-9. Threshold gate. to as an inhibiting signal since its 

function is to prevent the gate 
from opening. A similar modification of an and gate produces an 
equivalent not-and circuit. 

Another type of controlled gate appears in Fig. 3-9, where we can 
recognize the two input diodes as an and gate. T)\ and D 2 normally 
conduct, maintaining the voltage of point a at zero, provided that at 
least one input remains zero. C charges to the negative control voltage 
e c . To prevent this negative voltage also appearing at the output, we 
decouple through the diode D . 

After opening the and gate (large positive pulses at all inputs), what- 
ever signal appears at point a must be transmitted by C to point b. If 
e c is a slightly negative d-c bias, D„ will now be forward-biased, allowing 
the output to rise to the difference between the minimum applied input 



ei° K 



D 2 
e 2 ° K- 



Sec. 3-4] 



DIODE GATES 



77 



pulse and the control voltage. When ei is that particular pulse, the out- 
put becomes 

(3-12) 



ei 



e c >0 



where e c , the initial charge on the coupling capacitor, may be considered 
as the transmission threshold. Coincidences of input signals of greater 
amplitude are transmitted, while those of lesser amplitude cannot establish 
a transmission path and will not produce any output [Eq. (3-12)]. If 
the threshold is set slightly above the noise level, then the random noise 
pulses will not be transmitted and will not register falsely at the output. 



f 



12345678 



nnnnnnnn 



nnnnn 





t 




/ 

/ 


\ 
\ 


'c 




1 

1 


X 

N. 



/inrih 



4 5 6 7 
Fig. 3-10. Controlled-gate input-output pulses. 

An alternative mode of gate operation occurs when we initially set e c 
negative enough to prevent transmission for all possible amplitude input 
pulses. Only after a positive control signal e c charges the capacitor 
to a voltage close to zero will satisfaction of the and gate produce an 
output. This switching action is similar to that discussed for the first 
controlled gate considered. But where the first gate was primed immedi- 
ately upon the injection of the control pulse, this gate is not ready for 
transmission until considerably later, until C charges. Even though 
one of the input diodes always conducts, the charge and discharge time 
constants R C C are quite long. In the pulse-time relationships (Fig. 3-10) , 
note that the direct consequence of this is very poor transmission of 
pulse 4, good transmission of pulse 5, proper transmission of pulse 6, and 
false transmission of pulse 7. Obviously, use of this gate must be limited 
to pulse trains having wide separation. Replacing B c with a diode con- 
nected for fast charging of C during the positive control pulse improves 
the turn-on time, but not the gate's final recovery time. 



78 



MODELS AND SHAPING 



[Chap. 3 



3-5. Diode Arrays. One problem which arises quite often when 
information must be transmitted along several channels is how to select 
the proper one out of a number of possibilities. But the problem may 
be even more complicated; the same input may have to be transferred 
between various paths in a controlled time sequence. We can do 
this by using a number of independent diode gates. One input to each 
gate is used for the information to be transmitted while the other 
inputs are used for the control signals, i.e., the pulses that open and 
close the gate at the appropriate times. 



Control pulses 
bed 



«\ 




k. 




X 




X 






X 



X 



X 



k. 



X 



-o6 



Fig. 3-11. and array used for transmission selection. 



.*^. 



X 



-°1 



-°2 



Outputs 
°4 



X 



-°5 



X 



When these gates are combined into one matrix, we refer to the 
resultant circuit as a diode array. Figure 3-11 illustrates one such array 
where the excitation of two out of four control channels opens an infor- 
mation path for negative signals. The diodes connected to any one of the 
information channels (1 to 6) constitute an and gate. As they are 
normally conducting through the control-pulse generator, they establish a 
low-impedance path, shunting the signal to ground. However, if large 
negative pulses are simultaneously applied to two out of the four control 
channels (a, 6, c, and d), the associated pair of diodes are back-biased, 
the gate is opened, and transmission along one path becomes possible. 



Sec. 3-6] diode gates 79 

For example, the excitation of a and b opens path 1, the excitation 
of a and c opens path 2, etc. 

The statement of the particular two-out-of-four coding used for the 
selection of one out of six channels may be presented in a tabular form, 
called a truth table. We indicate by the symbol 1 that a pulse must be 
present in a particular control channel and by the symbol that it may be 
absent. The array of Fig. 3-11 satisfies the following statement: 







Pulse 


inputs 




Channel 










a 


b 


c 


d 


1 


1 


1 








2 


1 





1 





3 


1 








1 


4 





1 


1 





5 





1 





1 


6 








1 


1 



If all signal sources in the various channels (1,2,..., 6) are identical, 
e.g., a constant negative voltage E cc , then upon the application of the 
appropriate control signals, an output pulse will appear in a single 
channel. Thus the array translates the two pulses ab to mean number 1, 
as shown in the truth table. The simultaneous excitation of two out of 
four inputs steers the output to one point out of the six possibilities. 

More complex logic situations may be satisfied by adding additional 
diodes in a larger matrix. These may appear as ors as well as ands, 
and if both positive and negative control pulses are available, not 
diodes can also be included. 

3-6. Diode Bridge— Steady-state Response. Under circumstances 
where the amplitude of the output contains important information, the 
gate requirements become extremely critical. Neither of the circuits 
examined in Sec. 3-4 is satisfactory for amplitude gating; in both the 
output voltage not only varies as a function of the control signal, but 
also contains some percentage of the control signal, even when all inputs 
are zero. For example, in the gate of Fig. 3-8 a small negative voltage 
is developed across the forward resistance of the output diode as a direct 
consequence of the current flow to E cc . 

One convenient method of avoiding these errors is by using a balanced 
system driven by two equal signals of opposite polarity. The error 
voltage produced by each signal will be equal and opposite, canceling out. 
Since we know that in linear circuits almost complete voltage cancella- 
tion takes place in a balanced bridge, wj might well consider the appli- 



80 



MODELS AND SHAPING 



[Chap. 3 



cation of a diode bridge for amplitude gating (Fig. 3-12). For the output 
to be completely free of the control signal, i.e., not contain any component 
directly contributed by this signal, the bridge must remain balanced 
under all conditions. Or rather, we must limit circuit operation to the 
conditions that will not unbalance the bridge. 
Two balance conditions are obvious, the first when all diodes conduct : 






111 



(3-13) 



and the second when all are back-biased : 






Tri 



(3-14) 




To satisfy both Eqs. (3-13) and (3-14), the same type of diode is not 
only used for each arm of the bridge but is even individually selected and 
matched so that all will have identical forward and reverse resistance. 

Sometimes to further improve the 
balance, additional series and shunt 
resistors are added to each diode. 

Other conditions of bridge balance 
are also possible, for example, D x 
and Z) 2 on and D 3 and Z>4 off or vice 
versa. We shall see below that this 
represents an impossible mode of 
bridge behavior and need not be con- 
sidered. The remaining possible 
combinations of diode states occur 
when one set of diagonal diodes, Di 
and Dt,, conducts and the other set, 
D 2 and D 3 , is back-biased. But this leaves us with an unbalanced bridge, 
an undesirable situation which must be avoided. 

From the above discussion, we conclude that the sole function of the 
control or gating signal is the closing and opening of the gate, i.e., bring- 
ing the diodes into and out of conduction. Any large enough signal 
will cause switching, but to guarantee definitive diode states the signal 
should switch them smartly. For this reason we choose a symmetrical 
square wave, having a peak-to-peak amplitude of IE for our gating volt- 
age. If this signal is supplied from a source unbalanced with respect to 
the balance point of the bridge (in this case ground), then all diodes will 
not be equally excited, and with a very unbalanced drive some diodes may 
remain completely unexcited. We must therefore either use a balanced 



Fig. 3-12. Diode bridge. 




Sec. 3-6] diode gates , 81 

generator output or drive the bridge through a transformer. For the 
polarity indicated in Fig. 3-12, all diodes are conducting, and on reversed 
polarity, all become back-biased. However, this statement implicitly 
assumes that the contribution from e\, which will be examined below, is 
not large enough to turn off a conducting diode or to turn on a non- 
conducting one. 

When e„ forces all diodes into their active region, they present a very 
low resistance from the output to ground. By replacing them with their 
forward resistance, we see that across 
either diagonal ac or bd (Fig. 3-12), 
the resistance is the same as that 
of a single diode r f . Secondly, when 
the bridge is properly balanced, the 
voltage from 6 to d, due to e„, must be 
zero; it depends only on e\. Like- 
wise, the voltage developed from a 
to c depends only on e c . ?™- 3 : f 13 . Equjyatart bridge circuit- 

J the switch position is a function of the 

The reversal of the control signal control signal, 
back-biases all diodes, allowing us 

to replace them by their reverse resistance. The resistance from output 
to ground, bd, now becomes r r , and if the bridge is still properly balanced, 
again e c will not contribute to the output. 

Thus the gating signal causes sequential shunting of the output, first 
by r f and then by r r . The circuit related to ei appears similar to the 
shunt clipper of Chap. 2 where the clipping diode shunted the output 
by its forward resistance to prevent undesired transmission and by its 
reverse resistance to allow desired signal transmission. This leads us 
to interpret bridge operation as controlled clipping with its information 
transmission path a function of the control signal period (and voltage) 
rather than input signal amplitude. Since the two circuit states shown in 
Fig. 3-13 are the same as those existing in the clipper, the problem 
posed in solving for the optimum value of Ri must lead to the identical 
value found in Chap. 2 [Eq. (2-9)] : 

Ri = y/r T r f 

We shall now direct our attention to the input signal and examine the 
circumstances under which it could unbalance the bridge by changing the 
state of one or more diodes. A straightforward and convenient approach 
is to first calculate, and then compare, the two current components in 
the individual diode, one produced by the gating source and the other 
by the input signal. When the total diode current changes sign, the 
diode changes state. The individual diode current component maintain- 
ing the bridge closed (diodes conducting) is supplied by the gating signal. 



82 , MODELS AND SHAPING 

Solving the model of Fig. 3-14 yields 

E 



lea — 



2(i2. + r/) 



[Chap. 3 



(3-15) 



The maximum contribution from e\ occurs at the peak of the input 
voltage. 

El m 



/l» = 



2(Ki + r,) 



(3-16) 



The total current supplied by each signal is twice the current flow in the 
individual diode due to that source. 




6+ E -6 

Fig. 3-14. Bridge current flow during conduction (all bridge resistors are r/). 

In Fig. 3-14 the direction of the individual current components is 
indicated by the arrows. Note that the two components subtract in Di 
and Di but add in D s and D s . The former diode pair is in danger of 
being forced out of conduction on the positive input peaks, and the latter 
on the negative peaks. To prevent this, the contribution from the gating 
signal [Eq. (3-15)] must always be greater than the current supplied by 
the signal source [Eq. (3-16)]. By writing the required inequality, we 
arrive at one relationship between the three remaining unknowns E, 
E lm , and R.: 

Ely, 

Ri 



i>- 



(3-17) 



Equation (3-17) takes the form given since both Ri and R. are much 
greater than r t . 

For a second relationship, we naturally turn to the second bridge 
state, the open gate (all diodes back-biased). The individual diode 
current supplied by the gating voltage flows in the direction opposite 



Sec. 3-6] diode gates 

to the previously found /„, and its amplitude is 



/c» = 



-E 



2(B. + r r ) 



83 



(3-18) 



The current furnished from the signal source still can flow in the same 
direction as I^>, but its value changes to 



Ev 



2(Ri + r T ) 



(3-19) 



Under these circumstances the direction of current flow is such that on 
the positive inputs the two components add in Di and Z>4 and subtract in 
Di and Dz. Conditions established may drive one pair of diagonal diodes 
into conduction on the positive peak and the other pair on the negative 
peak of the input signal. Of course, this would again seriously unbalance 
the bridge, and to prevent this, we must also satisfy the second inequality 



E h 



E 



Ri + rr R. + rv 

But since R, 4C r P and Ri <K r r , this simply reduces to 

E lm < E (3-20) 

An additional complication arises because a diode's resistance is not 
constant in each region as depicted by the piecewise-linear model, but 
varies rather widely with its cur- 
rent. By taking the slope of the 
curve (Fig. 3-15), we see that 
the forward incremental resistance 
decreases from a relatively high 
value at small currents to a very 
low nearly constant resistance for 
large forward-current flow. The 
inverse incremental resistance in- 
creases with increasing reverse-cur- 
rent flow to some maximum, and 
then begins decreasing. 



100 ma ■ ■ 

SO ma 
-100 v -50 v 



-100 jta 



12345» 



«d 



Fig. 3-15. Diode characteristic. 



Whether the bridge is open or closed, the additional current contri- 
bution from the signal source increases current flow in one and decreases 
it in the opposing diagonal diode pair, thus decreasing the resistance of 
the first and increasing that of the second set of diodes. These opposing 
changes just compound the unbalancing of the bridge. We can minimize 
this second-order effect through setting the quiescent point at the center 
of the forward linear region of the diode and permitting only small current 
excursions about this point. Since E/2 appears across each back-biased 



84 



MODELS AND SHAPING 



[Chap. 3 



diode (R, <K r r ), this voltage should place the diode in the center of its 
reverse linear region for optimum bridge performance. 

Examination of Eqs. (3-15), (3-16), (3-18), and (3-19) indicates that 
in both regions, e c may be considered as setting the quiescent current 
and ei as causing the incremental current variations about this point. 
And as in all piecewise-linear circuits, when the peak incremental current 
exceeds the quiescent value, the circuit limits. We conclude from Eq. 
(3-15) that with any given peak driving signal E, we should choose R, 
to supply sufficient forward current to set the conducting bridge in its 
linear region. In addition, in order to maintain small-signal performance, 
E\ m must be limited to some small fraction of E [Eq. (3-20)]. The exact 




t 



nfintN 



mr^ 



Fig. 3-16. Bridge input-output signal relationships. 

percentage depends on the second-order variation of diode resistance and, 
of course, will vary with diode type. Generally, satisfactory results 
have been obtained on limiting the signal to 10 to 50 per cent of E. 

Waveshapes of the input (square-wave control signal superimposed) 
and the resultant gate output, illustrating the gate's operation, are shown 
in Fig. 3-16. Note that the output appears only during the negative half 
period of the gating signal when the bridge is nonconducting. The gating 
signal chops the input into a train of equally spaced varying-amplitude 
pulses. For this reason the bridge gate is often referred to as a chopper, 
or a bridge modulator. 

Many alternative circuit configurations are practicable, each solving 
a particular amplitude gating problem. The bridge can be interposed in 
series with the transmission path rather than in shunt. It can be used in 



DIODE GATES 



85 



ei 



3 





^S' 1 


3 




ixZj 


o 




o 


s 



o 
o 
o 

§ 

o 



Fig. 3-17. Bilateral transmission bridge. 



Sec. 3-7] 

a push-pull system or the single-ended one previously discussed, driven 
by direct current pulses, or even sinusoidal signals. The bridge might 
be used for bilateral as well as unilateral transmission, since its resistance 
is independent of signal direction. 
One feasible circuit, .illustrating 
many versatile attributes of the diode 
bridge, is shown here in Fig. 3-17. 
The bilateral transmission path (ei to 
c 2 and vice versa) is instituted by 
forward-biasing the diodes and inter- 
rupted by back-biasing them. 

3-7. Diode Bridge — Transient Response. The response of the diode 
bridge to the leading and trailing edges of the control signal determines 
the switching time and, consequently, the output waveshape. Moreover, 
because of the interaction within the gate, the input signal will likewise 
play a role in determining the response of the bridge to the gating signal. 
An exact analysis of the total problem becomes unreasonably tedious, but 
the approximate response, sufficient to delineate the limitations of the 
bridge, is not difficult to calculate. We divide the problem into two 
parts : first, we find the time required to turn the bridge on and off with the 
signal set at zero; second, we calculate the response of the bridge output 
with respect to the input, while the bridge alternately conducts and 
opens. 




Fig. 3-18. Bridge model for the calculation of turn-on, turn-off times. 



For calculation of the turn-on, turn-off time, we can use the model 
of Fig. 3-18 and, if the bridge is perfectly balanced, the parallel com- 
bination of Ri and C„ will not influence its response. Assume that the 
bridge is open (diodes off), on the verge of switching closed. Since 



86 MODELS AND SHAPING [CHAP. 3 

R. «: r r , the initial charge on each diode capacitor is —E/2, maintaining 
the diode cut off. 

At t = 0, e c changes polarity. Each capacitor starts charging toward 
the new steady-state voltage of +E/2 with a time constant found from 
the series-parallel diode branches, 

n = (r, || R.)C d ^ fi.C« (3-21) 

In calculating Eq. (3-21) we might note that, in terms of the effective 
resistance and capacity, the four diodes in the balanced bridge look 
like a single diode. Thus the exponential response of the voltage across 
a single diode becomes 

e d = yiE - Eer'i* (3-22) 

Eventually this voltage reaches zero and the diodes change state — they 
begin conducting. Equation (3-22), evaluated at e<» = 0, gives the bridge 
turn-off time as 

t x = ti In 2 (3-23) 

The now-active diodes reduce the circuit time constant to the almost 
insignificant t 2 . 

Ti S C d rf 

Cd continues charging, but toward a new steady-state value E it found 
by taking the Thevenin equivalent voltage across each diode in the model 
holding for this region. 

When the control signal again changes polarity, the energy stored 
in the shunting capacity maintains conduction for some short time. 
Instead of initially charging toward —E/2, each diode charges toward 

E *=-M. E 

with the time constant T2. 

The bridge turn-on time (diodes cut off) becomes 

tt = r 2 In 2 (3-24) 

which is the same fraction of the appropriate time constant rt as the 
turn-on time was of ti [compare Eqs. (3-23) and (3-24)]. On reaching 
zero after this very short time, the diode turns off and its capacity 
recharges to —E/2, with the reverse time constant n. Recovery is 
virtually complete within four time constants. Figure 3-19 illustrates 



Sec. 3-7] 



DIODE GATES 



87 



the behavior of a single diode, showing both its turn-on and turn-off 
times. 

When the gating period is sufficiently long, these switching times 
become insignificant. They do, however, set a definite lower limit to the- 
usable square-wave period. In addition, any capacitive unbalance causes 
unequal diode turn-on and turn-off times, unbalancing the bridge and 
producing sharp exponential pulses at the output. Sometimes additional 
trimmer capacitors are placed across each diode and adjusted for proper 
capacitive balance. This, of course, increases C<j, and consequently the 
time required to switch the diodes from on to off, and vice versa. 




Fig. 3-19. Single-diode voltage waveshape produced by e* 

The solution of the second part of the response problem, i.e., calculation 
of the output waveshape with respect to the input itself, becomes almost 
trivial once we draw the proper model. Since the bridge behaves 
as a controlled clipper, it might well be represented as two resistors 
and a switch controlled by e c , alternately connecting one and then the 
other (Fig. 3-20a). The circuit has two time constants: a long one for 
the rise t» and a short one for the decay T4. 



t, S «,(C. + &) r 4 S* r,«7. + C d ) 



(3-25) 



We can expect t% to be much longer than the bridge turn-off time con- 
stant r\ [Eq. (3-21)], as a consequence of the extra stray capacity at the 
output, and it will determine the fastest allowable gating signal. 

If we now attempt to find the actual bridge response by combining the 
response of each signal taken individually, we find ourselves facing a 
rather difficult problem. The calculation of turn-on and turn-off times 
had ignored the presence of the input signal, which will, during the turn- 
off interval, contribute additional current to one diagonal pair and 
reduce that flowing in the other pair. A rising voltage appears across the 
diodes because of the gating signal (Fig. 3-19) ; therefore Eq. (3-20) will 
not remain satisfied until the gate is completely off. The two diagonal 



88 



MODELS AND SHAPING 



[Chap. 3 



diodes D 2 and D% conduct first, unbalancing the bridge. A similar effect 
appears during the turn-on time; one pair of diagonal diodes cut off 
before the other pair, again causing bridge unbalance. 

Because of this unbalancing, exponential overshoots contributed by 
the gating signal are evidenced at the output (Fig. 3-20c). By recon- 
sidering the previous simplified discussion, the method of minimizing then 
becomes apparent. When the time constants n and t 2 are small, the 
bridge recovers fast, remaining unbalanced for the smallest possible time 
and producing only narrow overshoots. Excessive output capacity 
may also slow circuit response, preventing the rapid rise shown below. 



Ri 
9— MV 




f(ec) 






4*2 



\ f\ r 



(a) 



(b) 




Fig. 3-20. (a) Bridge input-output equivalent model; (b) ideal output waveshape 
(constant input voltage) ; (c) actual output response. 

However, these overshoots do improve the time response of the bridge, 
and if they are sufficiently narrow they may not disturb us at all. 

3-8. Concluding Remarks on Diode Gates. Most gates may be 
broken down into combinations of the various types discussed in this 
chapter. The methods of analysis used are applicable to all since the 
problems faced are similar: rise and decay times, steady-state response, 
and the time required to open and close the gate. 

Our discussion of the and and or gates purposely omitted the last 
point in order to simplify the first look at the subject. The student may 
now go back and with the material from Sec. 3-7 reexamine the earlier 
gates. In those gates the energy-storage elements across the diodes also 
prevent the immediate change of state; they maintain the previous condi- 



Sec. 3-8] diode gates 89 

tion in the face of circuit or excitation changes until they charge or 
discharge, as the case may be. 

In the first sections of this chapter, the time constants given are 
those which affect the response after the diode has changed state. With 
diodes in common use, the opening and closure times are too small to 
be significant except when we try gating extremely narrow pulses. Of 
course, any additional stray capacity will change the circuit behavior and 
increase the significance of the switching delay. 

Semiconductor diodes introduce additional delays in that they them- 
selves do not immediately switch from on to off and back again upon 
pulse excitation. As a consequence of their physical structure, they 
have three inherent delays: 

1. Turn-on time 

2. Turn-off time 

3. Storage time 

Turn-on and turn-off times correspond to the very short time required 
to inject and sweep out the majority charge carriers in the diode materials. 
They depend on the physical structure, diode material, junction area, 
and mean electrical path. Generally, these times are relatively insig- 
nificant in determining total response, as they are much less than the 
storage time. 

Some majority carriers are transported across the diode junction into 
the other material, where they become minority carriers; i.e., the current 
flow carries holes from the p to the n material and electrons from the 
n to the p material. Upon suddenly back-biasing the diode, forward 
current keeps flowing until recombination of the minority carriers is 
' virtually completed. This again depends on the physical structure of the 
diode and even upon its crystal lattice configuration. The storage time 
varies widely with diode type, from a few microseconds to several milli- 
microseconds. Data available from the manufacturer aid in selecting the 
proper diodes for the various waveshapes employed; obviously, fast pulses 
cannot be transmitted by a diode which does not respond until after the 
pulse has disappeared. 

At this point we shall briefly summarize our mode of attack on the 
nonlinear circuits analyzed in this chapter. A developmental procedure 
was followed, starting with the simplest possible circuit configuration and, 
step by step, increasing the complexity. In the first stage, we attempted 
to gain perspective by evaluating the ,steady-state operation and com- 
pletely ignoring the transition from the initial to the final conditions. 
Knowledge of the starting point and of the end point aids in mapping the 
proper path between them. Our second consideration was the effects of 
the external energy-storage elements on the system response to the 
drive signals. At this time simplification, even drastic simplification, 



90 



MODELS AND SHAPING 



[Chap. 3 



of the circuit avoids the necessity of concurrently treating more than a 
single independent energy-storage element. 

Our third step concerned itself with the external capacity affecting 
the switching response of the nonlinear circuit element. Again we cal- 
culated the response as if this were an independent problem and used 
the results primarily to establish inherent circuit limitations. Finally, 
we had to consider any additional complexity introduced by the circuit- 
element response: diode on, off, and storage time; possible resistor and 
capacity nonlinearity ; and any other phenomena which might affect 
the response in even minor ways. 



PROBLEMS 

8-1. In the ob gate of Fig. 3-la the three signal sources have relatively high imped- 
ance (2 K each) and therefore, by comparison, the diodes may be assumed to be ideal 
elements. The inputs are shown in Fig. 3-21. 

(a) What is the minimum load resistor which will ensure only one conducting diode 
at t = 0.3 sec? 

(b) If R = 10 K, at what time will D, begin conducting? At what time will Di 
cease conduction? 

(c) What is the effective load (e/i) on each of the three signal sources at a time 
midway between the two answers of part &? 

*2 



20 v 




0.2 0.7 sec 



15 v 




1 sec, 
Fig. 3-21 



, 


\h 






40 v 





















t 




Is 


ec 



3-2. The inputs of the or gate of Fig. 3-2 are each excited by a positive pulse: ei is 
a 4-^sec-wide 20-volt pulse injected at I = and en is a 30-volt 5-^sec pulse applied 
1 /usee later. Draw the output waveshape, labeling all important voltage values and 
time constants. The circuit parameters are given below. (Make all reasonable 
approximations.) 

r.i = r, 2 = 500 ohms R = 20 K 

d = C 2 = 10 md Co = 0.001 pi 

t, = 10 ohms r, = 100 K 




Fio. 3-22 



DIODE GATES 



91 



8-8. Assume that the diodes in the gate of Fig. 3-22 are ideal and that they are 
excited from low-impedance sources. 

(a) Sketch the output, giving all voltages and time constants if at t = a 1-msec 
15-volt positive pulse is applied at ei and a 2-msec 5-volt pulse at e». 

(b) Repeat part a if D» is reversed. 

8-4. (a) Sketch the output waveshape of the circuit of Fig. 3-23 if the diodes may 
be considered ideal. 

(6) Show that the time constant of the input coupling circuit must be long compared 

with the rise time of the gate and the pulse width for optimum circuit response 

(nonideal diode). 

t,usec „ 

♦ 0.01 /it 




Fig. 3-23 

8-6. (a) Draw the output waveshape of the and gate of Fig. 3-24 if both inputs 
are simultaneously excited by 50-/iSec pulses of -20 volts amplitude. Repeat if one 
pulse is reduced to —10 volts and the other remains at —20 volts. 

(6) Repeat part a for the circuit resulting when D a is removed. 

+25 v 



-°«o 




-30 v 



Fig. 3-24 



8-6. The and gate of Fig. 3-6 is excited by a unit step of 30 volts at ei and a stair- 
case voltage which increases in steps of 10 volts every 5 fisec at e 2 . A short time after 
the 40-volt step is reached, both inputs simultaneously drop to zero. The diode and 
circuit parameters are 



r, = 200 ohms 
r, = 500 K 
d =0 
r, = 500 ohms 



C. - 250 iml 
R - 20 K 
E - 25 volts 



Sketch the approximate output, indicating all time constants and important voltage 
values. 

8-7. (a) If En, > ei > e s > in an and gate and if the signal source impedance is 
very small, show that the gate's excitation will result in a small jump in the output 



92 



MODELS AND SHAPING 



[Chap. 3 



waveshape, after which it will rise exponentially to its final value. Calculate the 
value of this jump in terms of the circuit parameters. (Hint: Treat this gate as a 
compensated attenuator at t = 0.) 

(6) Will such a jump appear in the ok gate? If it will, how large is it? Explain 
your answer. 

3-8. The gate of Fig. 3-25 is connected as shown to two inputs, a pulse of 10 volts 
and 10 msec duration and a unit step of 100 volts. 

(a) Find and sketch e„, labeling the waveshape with respect to important voltages 
and times. 

(6) If a 1,000-ohm relay coil which will operate when the current reaches 1 ma is 
inserted in series with D 2 , at what time will the relay be energized? When will it be 
deenergized? 



10 v 




K 



Di 



9+25 v 
C10K 



t, msec- 



100 v-=- 



10 '! 

10 h 



-M- 



1 K< 



Fig. 3-25 




Fig. 3-26 



3-9. Figure 3-26 simulates the action of a single diode in a gate slowly recovering 
from a back-biased toward a conduction region. The switch is opened at t = 0, and 
the various circuit parameters are r c = 500 ohms, R = 20 K, C = 0.01 /rf, Ey, = 
50 volts, E a = 10 volts, and Et = 5 volts. 

(a) Plot the time response of this circuit. 

(6) Evaluate the slopes of the two exponential segments at the point where they 
meet. 

(c) Prove, in general terms, that in a circuit such as the one shown, the slopes of the 
two exponential segments are always equal at the point of intersection (r c <SC R). 

(d) Would the statement of part c be applicable if the diode was biased so as to 
conduct at a value other than zero? 



?+100v 



rioK 



200 K 
«i° — VA — W- 



Ci 



Fig. 3-27 



-tt-°«2 



:ik 



3-10. (a) Show that the counting circuit of Fig. 3-27 will have an output propor- 
tional to the number of applied input pulses (ei) provided that they are of equal 
amplitude and duration. Also discuss the possibility of erasing this stored informa- 
tion upon the injection of a sufficiently large negative pulse at e^. 



DIODE GATES 



93 



(6) If the amplitude of each pulse is 100 volts and its duration is only 2 ^sec, what 
size capacitor must we use so that the output will rise by approximately 1 volt/pulse 
for up to 10 pulses? Sketch the output waveshape. 

(c) Under the conditions of part b, how many pulses have been applied if the out- 
put reads slightly less than 25 volts? 

3-11. In the controlled gate shown in Fig. 3-28 the controlling pulse is a 1,000-jusec 
20-volt pulse which raises the normally negative control voltage from —20 volts to 

zero. 

(o) A train of 10-volt 20-jusee positive pulses, spaced 100 /isec apart (from the end 
of one pulse to the beginning of the next one is 100 /isec) is the input to ei. Sketch 
the output waveshape from slightly before the control pulse is applied until slightly 
after. 

(b) Repeat part a if we simultaneously apply a 25-volt positive pulse train of the 
same duration and period at c 2 . 

(c) Does this gate offer any advantage over a similar operating version of the one 
shown in Fig. 3-9? 




Fig. 3-28 



3-12. (a) Draw the circuit of a direct-coupled controlled and-not gate designed 
to operate on positive pulses (similar to the and-oh gate of Fig. 3-8 except that it 
should be direct-coupled to all signal sources). 

(b) Designate the relative amplitudes of the pulses which should be used. 

(c) Sketch the input and output waveshapes and compare the results with that 
obtained for the gate of Fig. 3-8. Does this circuit offer any advantages? 

3-13. In an automatic milling machine, the work must be properly positioned on 
the bed before the machine can be permitted to operate. The proper position is indi- 
cated when the states of three separate single-pole double-throw (SPDT) switches 
are changed by the pressure of the work. The center point of each switch is connected 
to a constant voltage (positive), and the other terminals are available for connection 
to various diode gates. 

(a) Sketch a circuit which will indicate when the machine can start functioning. 

(6) Sketch a circuit which can be used at the same time as the gate of part a and 
which will keep the positioning mechanism in operation until the work is properly 
seated. 

(c) Show how you would modify the gate of part b so that the output decreases by 
equal increments as each switch is activated in any sequence. 



94 



MODELS AND SHAPING 



[Chap. 3 



(<J) If you used diodes in the above parts, redraw the circuits so that they do not 
require any diodes. If you did not use diodes, repeat using them as elements in the 
various gates. 

3-14. Figure 3-29 illustrates a simple controlled amplitude gate which will operate 
for positive or negative input signals. 

(a) If ei is a 20-volt peak-to-peak sinusoidal signal, how large must we make the 
peak-to-peak square-wave control signal so that the diodes will not change state as a 
function of ei but only as a function of e c 1 Specify your answer in terms of the circuit 
and diode parameters r„ ry, Ri, and R, (r r 58> Ri S> r/, r r 55> R, S> iy). 

(6) Specify the optimum value of Ri for the maximum ratio of desired to undesired 
output, when the peak-to-peak control signal is 100 volts (jy = 10 ohms, r, = 100 K, 
and R\ = R,). 

(c) Sketch the input and output waveshapes to scale, under the conditions of 
part 6, if the period of the control signal is 5 per cent of the sine-wave period. 

(d) Repeat part c when the input signal is zero. 



«i 



0-4- 
r, 4 



R,< 



-oe. 



6 e ° ! 

Fio. 3-29 

3-16. Show the circuit of a diode array that will satisfy the following three condi- 
tions : 

1. Select the proper 1 out of 10 channels upon the simultaneous application of pulses 
in 3 out of 5 channels. 

2. Reject all odd outputs upon the application of an inhibit pulse. 

3. Reject all even outputs upon the application of a second inhibit pulse. 

3-16. Construct a diode array satisfying the following truth table. All channels 
marked with an asterisk are statements of an ok gate; all others are and gates. Under 
which conditions are there multiple outputs? 



Channel 


a 


6 


c 


d 


e 


1 








1 


1 


1 


2 










1 


1 


3 









1 





4* 






1 








5 






1 








6* 












1 


7 









1 


1 


8* 













1 



3-17. Evaluate the percentage of the control signal appearing at the output of a 
diode bridge if one diode, D\, differs by 5 per cent from the other diodes. Its parame- 
ters are r' r — 1.05r r and r t = 0.95r/ expressed in terms of their nominal values 



DIODE GATES 95 

Three cases should be considered: 

(a) R. = 0.2 \Avv- 

(b) R. =■ Vw 

(c) iJ, = 5 y/r,r f . 

In all cases Ri remains fixed at ^/r r r/. Can you draw any general conclusions as to 
the best choice of R, in the face of the expected small variations in diode parameters? 
(Let r/ = 10 ohms and r r = 100 K in your calculations.) 

8-18. (o) Find the optimum reflected impedance through each transformer to the 
common bridge circuit for the maximum ratio of desired to undesired signal-voltage 
transmission (Fig. 3-17). What attenuation factor does the bridge introduce under 
these circumstances? 

(6) Further show that the conditions found in part o will also apply to a shunt con- 
nection. (The bridge is connected across the two transformer windings, which are 
in parallel.) 

3-19. Diodes having the characteristics of Fig. 3-15 are used in the bridge of Fig. 
3-12. The maximum reverse diode voltage is 100 volts, and for safety's sake we limit 
our driving signal to 75 per cent of the absolute maximum. We desire that the diode 
should be biased at 1.5 volts when it is in its active region. Furthermore, the peak 
input signal will never- exceed 20 volts, thus ensuring that the bridge remains balanced 
at all times. Find the values of R, and Ri for optimum operation under the above 
conditions. 

3-20. A diode bridge having the parameters listed below is to be controlled by a 
square wave, and a total of not more than 5 per cent of the period can be devoted to 
the turn-on and turn-off transients. What is the fastest allowable gating signal? 

r t = 100 ohms Ri = s/r r r/ 
r, = 250 K R. = 50 K 

C d = 10 wf C„ = 250 /uif 

8-21. In a diode bridge circuit any capacitive unbalance will cause large spikes to 
appear at the output. If R, is small, calculate the time constant and relative ampli- 
tude of these spikes if Cn is (o) twice all other diode capacity; (b) one-half all other 
diode capacity. Take C„ as zero in this problem. (Hint: Treat in the same manner 
as the compensated attenuator of Chap. 1 and use a constant input signal.) 



BIBLIOGRAPHY 

Brown, D. R., and N. Rochester: Rectifier Networks for Multiposition Switching, 

Proc. IRE, vol. 37, no. 2, pp. 139-147, 1949. 
Chen, T. C. : Digital Computer Coincidence and Mixing Circuits, Proc. IRE, vol. 38, 

no. 5, pp. 511-514, 1950. 
Hussey, L. W.: Semiconductor Diode Gates, Bell System Tech. J., vol. 32, pp. 1137- 

1154, September, 1953. 
Millman, J., and T. H. Puckett: Accurate Linear Bidirectional Diode Gates, Proc. 

IRE, vol. 43, no. 1, pp. 27-37, 1955. 
and H. Taub: "Pulse and Digital Circuits," McGraw-Hill Book Company, 

Inc., New York, 1956. 
Richards, R. IT.: "Digital Computer Components and Circuits," D. Van Nostrand 

Company, Inc., Princeton, N.J., 1957. 



CHAPTER 4 

SIMPLE TRIODE, TRANSISTOR, AND PENTODE 
MODELS AND CIRCUITS 



An active element such as a triode, pentode, or transistor, with its 
unidirectional transmission path, its internal amplification, and related 
conversion of energy from the power source to the signal output, adds 
a third dimension to our capabilities for wave generation and shaping. 
Where previously we had only linear and nonlinear elements at our dis- 
posal, in this chapter we shall discuss some simple uses for a controlled 
energy source which can be incorporated within, instead of remaining 
external to, the wave-forming network. 

4-1. Triode Models. Consider, for example, the terminal character- 
istics of a typical average triode, expressed by the two experimentally 
determined families of curves shown in Fig. 4-1. One set represents the 
input grid characteristics, and the other, the plate volt-ampere (output- 
terminal) response. Any specific tube of the same type may be expected 
to deviate by up to ± 30 per cent or even more from the average character- 
istics given by the manufacturer, as a consequence of the manufacturing 
tolerances. Thus we should expect a macroscopic model to represent 
reasonably well, but not necessarily exactly, the conditions expressed in 
Fig. 4-1. 

For the simplest analysis of the subsequent circuits, the model should 
be a piecewise-linear one, allowing us to write a separate linear defining 
equation for each region of operation. In formulating the model, we 
first observe that the plate voltage has no effect on the grid characteristics 
in the negative grid region and but small effect when the grid voltage 
becomes slightly positive. If this second-order phenomenon is ignored 
by using the average of the curves shown to define the complete grid 
response (Fig. 4-la), then the approximate characteristic is identical 
with that given for a diode (Chap. 2) and we can represent the grid- 
cathode behavior by an equivalent diode (Fig. 4-3). This diode's 
forward resistance r c depends on the particular tube type and will gen- 
erally lie between 300 and 2,500 ohms; for the tube shown, it is approxi- 
mately 1,400 ohms. The inverse resistance is so very large that it is 
almost always assumed to be infinite. 



Sec. 4-1] 



SIMPLE TKIODE AND TRANSISTOR CIRCUITS 



97 



Turning now to Fig. 4-16, the location of any individual plate character- 
istic, which happens to be of the same basic shape as the diode volt- 
ampere curve, is a function of the particular value of the applied grid-to- 
cathode voltage. All these curves are quite similar, one to the other, 
roughly parallel, and approximately equally spaced for equal grid-voltage 



50 V 
100 
150 




5 10 15 
e c , volts 



30 



120 



10 



/ / 






4? 










> 




















/ 




































' 




































1 


















/> 






















































/ J 






















^> 




































/' 














\ 


























* 












































7 






























fi 




/ 
































4 


i 


'/ y 
































.•*> 





































100 



400 



200 300 

Rate voltage e b , volts 
(b) 
Fig. 4-1. Typical triode characteristics of a 12AU7. (a) Grid volt-ampere terminal 
response; (6) plate volt-ampere characteristic. 



increments. We shall therefore extend the piecewise-linear concept as 
previously applied to the diode and approximate the totality of plate 
characteristics by a family of parallel, equally spaced straight lines 
(Fig. 4-2). The slope of these lines should be chosen for the best match 
to the original characteristics and will be roughly the reciprocal of the 
plate resistance averaged over the entire plane,' l/r p . Since at any con- 
stant value of plate current the ratio of the change in plate voltage to 



98 MODELS AND SHAPING 

the change in grid voltage is 

AeJ 



M = — 



Ac 



c it*™ constant 



[Chap. 4 
(4-1) 



the separation between the adjacent plate curves will be — p Ae„. 

From Fig. 4-2 we can see that a reasonable representation of a 12AU7 
requires an average n = 18 and r, = 8K With some other triodes the 
H may vary from 10 to 100 and the r p from 1 to 100 K. Usually the 
low values of r v are associated with the low-/* tubes. 



i e h "kee 




50 100 150 200 250 300 350 400 450 500 550 600 
[«-//Ae c *| Plate voltage e b , volts 

Flo. 4-2. Piecewise-linear approximation of the plate volt-ampere characteristics for 
type 12AU7 triode. 

In the interests of clarity, the derived linear characteristics are pre- 
sented in a separate diagram (Fig. 4-2) rather than superimposed on the 
actual curves. However, in order to find the piecewise-linear parameters 
used to classify the particular tube type (n, r p , r c ), we would have to 
refer to the manufacturer's characteristics, draw the family of straight 
lines, and then calculate these parameters. After this has been done 
once, there is no need to repeat the process whenever we must solve a 
new problem using the same tube. Even when the piecewise-linear 
characteristics are carefully approximated by eye, the deviation between 
the actual and the linear curves will usually be less than 10 per cent over 
the major portion of the region we are attempting to define, i.e., the first 
quadrant. As expected, the largest errors occur at low plate current 
where the curvature becomes rather severe. 

Once the curvilinear representation is decided, we can derive a mathe- 
matical expression for the simplified curves drawn. The parametric 



Sec. 4-1] simple triode and transistor circuits 99 

equation describing the family of straight lines of Fig. 4-2 becomes 

e\> ~ — fie c + r p ih n > 0, e b > (4-2) 

where — iie e is the plate-voltage intercept. Equation (4-2) may be 
interpreted as representing the volt-ampere characteristics of a voltage 
generator — ne c having an internal impedance of r p (Fig. 4-3). As the 
control voltage changes by any increment, the open-circuited output 
changes proportionately and the linear curve moves to a new position 
parallel to itself. 

An additional limitation must also be imposed to restrict further the 
region defined by Eq. (4-2) : 

e b > ke e (4-3) 

when e e > 0. Unless the tube's operation is so restricted, the excessive 
grid current flow, with the consequent lowering of the value of r c , which 
occurs at low values of plate voltage and high grid voltage, may destroy 
the tube; the flimsy grid structure is not designed to dissipate the heat 
produced. In Eq. (4-3), the constant is a function of the interelectrode 
spacing and the physical construction of the tube and takes on a rela- 
tively small range of values; k usually lies between 0.5 and 2.5. If the 
tube is designed to operate under these conditions, then a new model is 
necessary to represent adequately the behavior at high positive grid 
voltages. The construction of such B D 

a model will be deferred until needed 
in Chap. 12. 

It follows, from the previous dis- 
cussion, that the macroscopic piece- 
wise-linear model of the triode is the 
one shown in Fig. 4-3. The two 

diodes serve" to limit operation to ' r 

the proper quadrants. The grid * 

diode permits grid current flow only FlG - 4_3 - Triode P^ewise-linear model. 
when the grid voltage is positive apd the plate diode prevents reverse 
plate current flow. 

When constructing the model of Fig. 4-3, we found it necessary to 
introduce a new element, a controlled source, in order to account for 
the influence of grid voltage over the plate current. The particular 
signal source chosen, M e c , is a voltage generator whose amplitude is a 
linear function of the grid-to-cathode controlling voltage. It reflects the 
input signal into the output loop and thus represents the voltage ampli- 
fication inherent in the tube. 

Under the normal operating conditions the grid diode remains back- 
biased while the plate diode conducts. We say that the tube "saturates ' ' 



«6 



100 MODELS AND SHAPING [CHAP. 4 

when the grid diode changes stage and conducts at e ok = e c = 0. Cutoff 
corresponds to back-biasing the plate diode (4 = 0), and by writing the 
voltage across this diode, the necessary condition becomes 

Ec = E cMot , < - ^ (4-4) 

where Em is the voltage appearing from plate to cathode when % = 0. 

If instead we choose to replace the plate circuit by its dual represen- 
tation (Norton equivalent) , the voltage-controlled voltage source becomes 
a voltage-controlled current generator. This transformation follows 
directly from Eq. (4-2), which is rewritten so that the plate current is the 
dependent rather than the independent variable. The new denning 
equation is 

ib = g m e e + — (4-5) 

r v 

where g m = y./r p is called the transconductance. Even though the out- 
a „ put-controlled source is now a cur- 

-M ? rent generator (g m e c ), its value is 



-W- 



still a function of the grid voltage. 

S m e c{^)\ ^ g p e b Furthermore, the transformation 

converted the series resistance into 

a shunt conductance (Fig. 4-4). 



Fig. 4-4. Alternative triode model. The twQ diodes uged in the model 

reflect the physical limitations of the tube, and therefore they will remain 
invariant across the change in modular representation. 

Triode Amplifier. As an example of how the piecewise-linear model 
is applied in the solution of specific circuits, we shall concern ourselves 
with the single-tube amplifier of Fig. 4-5a, which employs the tube whose 
characteristics are given in Figs. 4-1 and 4-2. The assumption is made 
that the driving signal is of a high enough frequency so that the cathode 
remains completely bypassed with respect to all signal components. 
Two problems require solution: first, the circuit's quiescent conditions 
and, second, the input-output transfer characteristic. 

At the quiescent point the tube is biased within its grid base, i.e., 
between cutoff and zero, and the grid is nonconducting. The grid-to- 
cathode voltage is determined by the current flow through the cathode 
resistor. 

E cg = -E K = -h q Rt (4-6) 

As shown in Fig. 4-5b, the substitution of the value of E cq , given 
by Eq. (4-6), into the model yields a generator whose terminal voltage 
is proportional to the current flow through it. But since such a response 
is the same as the voltage drop appearing across a resistor due to its 



Sec. 4-1] simple triode and transistor circuits 101 

current flow, this particular voltage source can be replaced by an equiv- 
alent 18-K resistor, /ii?3. The reduced quiescent circuit consists of the 
power supply in series with various resistors; therefore 



Ebi 



300 



69 R* + r p + ( M + 1)« 3 20 K + 8 K + (19)1 K 

From the circuit we see that the other quiescent conditions are 

E K = hgRz = 6.4 volts 

E 2q = E bb - hcRz = 172 volts 

where h 9 is as given in Eq. (4-7). 



S 6.4 ma (4-7) 



(4-8) 



«iW 




(a) 



(b) 



Fig. 4-5. (a) Triode amplifier; (6) piecewise-linear model. 

Before beginning the evaluation of the transfer characteristic, we might 
observe that the linearization of the tube characteristics results in three 
linear regions of operation. Furthermore, the segments of the transfer 
curve for each region are contiguous, permitting their solution by zone 
superposition, i.e., the calculation of the transfer slope within each region 
and the location of the boundary-point coordinates. These individual 
segments can then be superimposed on the input-output plane with 
respect to the quiescent point to obtain the over-all response. 

Since only the slope is needed, it is only necessary to define an operating 
region and to consider the incremental output response with respect 
to some variation of the input signal falling within the defined zone. In 
this way the complete output variation is contributed by the controlled 
source and the' effects of the power supplies may be ignored. The three 
operating regions are: 

1. The large positive peaks of the input signal force the tube into grid 
conduction over some function of the cycle (saturation). 



102 



MODELS AND SHAPING 



[Chap. 4 



2. The tube operates within its normal grid base. 

3. The negative peaks of the input signal are large enough to drive 
the tube into cutoff. 

Incremental models, which hold for each of these regions, may be 
derived from the piecewise-linear model of Fig. 4-56 by shorting all 
batteries and conducting diodes and opening any branches containing 
back-biased diodes. The three models appear in Fig. 4-6o to c. When 
considering the bounds of these regions, we would have to refer back 




<c> 

Fio. 4-6. Incremental models for the circuit of Fig. 4-5. (a) Saturation region; (6) 
active region; (c) cutoff region. 

to the complete model of Fig. 4-5b, where the complete bypassing of R 3 , 
afforded by Cz, permits its replacement by the constant voltage Ek 
[Eqs. (4-8)]. 

Region 1 : ei(t) > Ek (saturation) 

In this region, since Ri J5> r c , 

*** = -r» Ae i = rA Ae » = 0123 Aei 

r e + R. 11.4 

and from the incremental model of Fig. 4-6o, 

nRz 



Aet = — 



r, + Ri 
where the grid-tc-p'late amplification is 



Ae e = 0.123 A Aei 



(4-9) 



A = 



Ae2 
Ae„ 



uRt 



r T + R» 



18 X 20 K 
8 K + 20 K 



= -12.9 



Sec. 4-1] 



SIMPLE TBIODE AND TRANSISTOR CIRCUITS 



103 



Note that the conducting grid introduces an additional attenuation factor 
due to its loading of the signal-source impedance. The output voltage 
corresponding to the boundary value of the input may be found from 
Fig. 4-56 by setting e« = and replacing R 3 by the battery E K . The 
controlled source disappears at this point, and the simple resistive net- 
work of Pig. 4-7o yields 



Et, = Ek + It,r p 



„ En — E K 

&K + , r, ?„ 

Tp + «2 



^ 90 volts 



(4-10) 



where h, is the saturation value of plate current. 



Region 2: 



E K > e,(<) >E K - 



Eu 



Ek 



(active) 



The grid now operates within its 
normal base, it is nonconducting, and 
it does not load the signal source. 
Thus, when Ri ^> R„ Ae cl = Aei, and 

Ae * = ~ , if p Ae ° = A Ae ' (4" 11 ) 
r P + Ri 

As the negative portion of the signal 
cuts off the tube, the circuit enters 
the third region of operation, and by 
again substituting the battery Ek for 
Rz in the model of Fig. 4-56, we find 
the cutoff value of ei(t). The resultant 
simplified circuit appears in Fig. 4-76. 
Using Eq. (4-4), E„ and the equivalent 
input voltage become 

Em, 



E„ = e u (t) - E K < 



eu(t) < +E K 



Em — Ek 




(a) 

Fig. 4-7. limiting models for the 
amplifier of Fig. 4-5o. (a) Tube 
under saturation conditions; (6) 
tube cut off. 



Er 



- 16.3 volts 



-9.9 volts 



(4-12) 



The plate voltage at cutoff is simply Em- 
Region 3: e^t) < +E K - Ehb ~ Ek 



(i.e., below cutoff) 



Since the cutoff tube opens the transmission path for the input signal, 
the output will remain constant at E a and, in this region, the slope of 
the transfer characteristic is identically zero. 

The conclusion which may be drawn from the above calculations is 
that it is much easier to locate the circuit's break points than it is to find 



104 



MODELS AND SHAPING 



[Chap. 4 



the quiescent operating condition. In the triode, one coordinate of each 
of these limits is already known; in one case it is where the plate current 
drops to zero, and in the other it is where the grid-to-cathode voltage rises 
to zero. We can always simplify the piecewise-linear models at these two 
boundaries by eliminating the unessentials : the controlled source is 
shorted at e c = 0, and the complete plate loop is opened at 4 = 0. 
Finally, the remaining extremely simple network is solved for the single 
unknown coordinate. 



Slope=0 




-9.9v 6.4 v erft) 

Fig. 4-8. Transfer characteristic of the one-tube amplifier of Fig. 4-5. 

Figure 4-8 illustrates the complete piecewise-linear transfer relation- 
ship, and as a summary of the previous discussion, all slopes and inter- 
cepts are labeled. The superimposed curve (dashed lines) is the transfer 
characteristic as might be evaluated point by point from the manu- 
facturer's tube characteristics. Agreement between the two curves is 
generally satisfactory, except in the vicinity of the break points, where 
the pronounced curvature of the actual plate characteristics results in 
wide divergence from the model representation. The slope of Fig. 4-8 
is the value of the incremental gain at that particular value of drive 
voltage. - If we are interested in linear amplification, the drive must be 
restricted to keep the operating locus in the active region. Furthermore, 
for the largest possible dynamic range, the Q point should be placed 
halfway between grid conduction and cutoff. 

The limitations as well as the advantages of the piecewise-linear 
analysis become apparent once we consider how to obtain the actual 
operating path of the single-stage amplifier of Fig. 4-5. 

In the graphical construction of this locus we must first locate the Q 
point, which lies along the d-c load line 



e h = En - i b (R 2 + R 3 ) 



(4-13) 



The line defined by Eq. (4-13) appears superimposed on the triode char- 
acteristics in Fig. 4-9. Secondly, the actual bias point is also determined 



105 



Sec. 4-1] simple tbiode and tbansistob cibcuits 

by a graphical construction. The bias equation (4-6) is cross-plotted 
on the plate characteristics, and its intersection with the d-c load line 
located. Finally, the locus of the time-varying output, the so-called 
a-c load line having the slope — l/Ri, is drawn through the Q point. 




200 300 

e b , volts 

Fig. 4-9. Graphical solution of the single-tube triode amplifier (the position of the a-c 
load line is exaggerated for the purposes of the illustration; it does not correspond to 
the circuit of Fig. 4-5). 



9+300 v 



12AX7 



/* = 100 
rp-70 K 
?•<,= ! K 





Fig. 4-10. Cathode follower for Example 4-1. (a) Circuit; (fc) equivalent piecewise- 
linear model. 

For simple circuits, a graphical construction is almost as easy as the 
analytical solution. In more complex circuits this is no longer true. 
Even in the single-tube amplifier, once the grid becomes positive we 
would have to solve two sets of graphically presented characteristics: 
first, those of the grid to find the actual value of e c and, next, those of the 
plate circuit to find the corresponding output voltage. 

Example 4-1. This example will consider the cathode follower of Fig. 4-10. We 
shall solve for the cathode resistor R K that places the quiescent point exactly in the 
center of the active region. 



106 MODELS AND SHAPING [CHAP. 4 

Solution. In the cutoff region the full supply voltage appears across the tube. 
Thus the grid-to-cathode voltage at the boundary is given by 

„ _ -500 . 

H'ckc = = — a volts 

M 

Since the net current flow is zero, the output voltage is Ev> = —200 volts, and the 
corresponding value of the input becomes 

Ei„ = E cU + Eu, 205 volts 

The saturation limits may be determined by setting e„ = in the model of Fig. 
4-106. By doing so, the controlled source is eliminated and 

/,- 50 ° 



r P +R K 

The output voltage, which is now equal to the input, is given by 
Ei, - E u - I b Jt K - 200 volts 

To obtain a second equation involving R K and h, we consider the quiescent condi- 
tions. When Ei — 0, the grid-to-cathode voltage may be expressed as 

Ed, - —Ett = 200 - R K It, 

Substitution of this equation into the network of Fig. 4-106, by way of the controlled 
source, requires the multiplication of each term by it. At the Q point the controlled 
source may be replaced by an equivalent resistor of Rk in series with a voltage source 
of — (m + 1)E CC volts. The tube's current, found from the single loop remaining after 
the substitution, is 

, Eg- (jx + V)E CC _ Ef. 

'' r P + ( Ji + 1)R K = R K 

where the approximation holds for a high-/* tube [Eu, <SC —E cc (ji + 1) and R K (fi + 1) » 
»■,]. In order for the quiescent point to lie in the center of the active region h, = h./2. 



Substituting values and solving results in Rk = 280 K. The previously unknown 
values can now be evaluated. 

lb, = 0.715 ma 

Ei, = +2.5 volts (from the more exact solution for h,) 
Ei. = Eu - 200 volts 

The piecewise-linear model has not only freed us, in our calculations 
of the tube quiescent point of operation, from any dependency on graph- 
ical construction, but also has contained within the model a diagrammatic 
representation of the complete range of tube behavior. All incremental 
models were directly derived from the single piecewise-linear model, 
and the particular constraints imposed on the regional models also 
explicitly appear. If, in any region of operation, a more exact calcula- 
tion as to the incremental behavior is needed, we can always return to 



Sec. 4-2] simple triode and transistor circuits 107 

the actual tube characteristics and calculate the incremental values of 
r p and m for the microscopic portion of the volt-ampere plane under 
review. This is particularly necessary when operating at low plate 
currents where the low slopes of the actual characteristics indicate a 
large increase in r r . In this same region, adjacent curves begin crowding 
together, with a consequent decrease in the incremental ju (Fig. 4-16). 

The piecewise-linear model cannot account for these variations in 
tube parameters since the basis of its construction was the linearization 
of the tube characteristics. Over most of the plane, the values of /i and 
r p are not much different from the constant values assumed in the linear 
curves of Fig. 4-2. We may conclude that the macroscopic model is 
generally valid for large-signal applications, that it is reasonably valid 
for small-signal analysis over much of the operating region, and that it 
represents a convenient first-order approximation over the remainder 
of the plane. 

4-2. Triode Clipping and Clamping Circuits. From a study of the 
transfer characteristic of Fig. 4-8 we conclude that, besides functioning 
as a linear amplifier within its active region, the nonlinearity of the 
triode may also be utilized as a double-ended clipper. Each break 
point corresponds to a change in state of one of the diodes used in the 
piecewise-linear representation (Fig. 4-3); the clipping of the positive 
input peak occurs as the grid is driven into conduction, and the negative 
peak is clipped in the plate circuit as the controlled source cuts off the 
tube. We generally reflect the cutoff action back to the grid and speak 
of the spacing between the two break points, in terms of the grid-to- 
cathode voltage, as the grid base of the particular tube. Control over 
the width of the transmission region is rather awkwardly effected; either 
the cutoff point must be changed by adjusting the plate supply voltage 
or we must replace the tube with one which will operate within a smaller 
or a larger grid base (i.e., a different value of m). 

In spite of this limitation, the triode is widely used as a clipper because 
advantage may be taken of its internal amplification to develop a large 
clipped output from a relatively small driving signal. For example, 
if a sinusoid having a peak-to-peak amplitude much larger than the grid 
base is chosen as the input, then the clipped output will be a fair approxi- 
mation of a square wave. Even better squaring results when a second 
stage is cascaded for further clipping and amplification. Simple differ- 
entiation converts the square wave into a pulse train which might be 
applied to other circuits for synchronization or control. 

The undesirable information transmission occurring in the saturation 
region may be minimized by inserting a very large resistor in series with 
the control grid. But since the input capacity of a triode is quite large, 
50 to 300 ppi, the resultant long input time constant adversely affects 




108 MODELS AND SHAPING [CHAP. 4 

the rise time of the stage. For this reason, clipping is usually limited 
to the cutoff region and a compound circuit, such as the cathode-coupled 
amplifier shown in Fig. 4-11, would be used instead for double-ended 
operation. Here each clipping region corresponds to one of the tubes 
being driven into cutoff, and therefore no special provision need be 
made for operation in the saturation region. 

Qualitatively, this circuit operates in the following manner: tube Ti 
is a cathode-follower input stage, injecting the input signal into the 

cathode circuit of T 2 . The negative 
bias E ee serves to set the proper 
quiescent conditions. At very large 
positive values of e-i, the positive 
common-cathode voltage cuts off tube 
T 2 , and through doing so destroys 
the transmission path from the input 
to the output. In this region the 
slope of the transfer characteristic 
becomes zero. As the input signal 
falls, Ti enters its active region and 
Fig. 4-11. Double-triode clipping cir- the circuit functions as a cathode- 
cuit— cathode-coupled amplifier. coupled amplifier. Linear ampli- 

fication ceases when the falling signal 
voltage finally cuts off T\, again interrupting the transmission path and 
again making the transfer slope zero. 

When the input is large enough to cut jT 2 off, the model of Fig. 4-12o 
holds and will be used to calculate one of the boundary values. We must 
first write the following grid-circuit defining equation: 

e c i = ei — itiRi — E ce (4-14) 

Substitution of Eq. (4-14) into the model of Fig. 4-12a results in the 
replacement of the controlled source ne c i by two voltage sources —nE cc 
and nei and an equivalent resistor fiR 3 . Thus the effect of the driving 
signal is brought to the fore and the plate current becomes 

. = E» — (ft + 1)E CC + m^i 
r, + (m + l)B t 

But in order to ensure that T 2 is off and that our argument as to the 
circuit operation is valid, the grid-to-cathode voltage of T 2 must be 
equal to or below cutoff. By taking the external loop voltages, from the 
grid to the cathode, we obtain 

e« = -E cc -E t <- Ebb " Ec ° ~ Es (4-15) 



Sec. 4-2] simple triode and teansistoe ciecuits 109 

where E% = %\Rz and the plate-to-cathode voltage of T 2 is Eu> — E cc — E 3 - 
Solving Eq. (4-15) for E t and equating to i b iR 3 yields 



E% 



E» - (ji + l)E cc _ En + nEn - 0* + l)E a 



M+ 1 



r p + ( M + l)fi, 



Rs 



The upper bounding value of the drive signal En is readily found by 
solving the above equation. 

The location of the other break point would be found in a similar 
manner by directing our attention to the model of Fig. 4-126. Again 
the additional information needed for a complete solution is furnished 
by the cutoff equation of the nonconducting tube T x . The two equations 
involving E% are 



Ez 



e c i 



ibiR3 — 



E» - 0» + l)E c , 



R, 



Ei 



r P + fl 2 + (/i + 1)R, 

Em, — E cc — E% 



Ecc — Ei < 



The simultaneous solution of these equations for En, the minimum value 
of the input which can be tolerated before Ti is driven into cutoff, com- 
pletes the solution of this problem. 

Triode-gr id-circuit Clamping. A 
further circuit application of the 
triode, with its two-diode repre- 
sentation (Fig. 4-3), is as a clamp. 
Both the grid and plate circuits 
will individually clamp an external 
signal in exactly the same manner 
as the diode clamps of Chap. 2, but 
because the equivalent diodes used 
in the triode model are irreversi- 
ble, only negative clamping is 
permitted. 

By properly adjusting the cath- 
ode bias resistor, the grid-circuit 
clamp level may be varied from zero to a positive value equal in magnitude 
to the tube's cutoff voltage. Since the triode grid has an almost infinite 
reverse resistance, this circuit would require a resistor, shunted from 
grid to cathode, in order to ensure the proper charging of the clamp 
capacitor (Sec. 2-5). 

This same clamping action is widely employed as a means of establish- 
ing the grid bias of the tube. In Fig. 4-13, the grid conduction on the 
positive peaks of the applied signal results in a current which charges C 
to the positive peak of the input voltage. Except for the short interval 




(a) 

Fig. 4-12. Piecewise models of the circuit 
of Fig. 4-11. (a) ?\ on and T 2 cut off; 
(ft) Ti on and Ti cut off. (Note: E cc < 0.) 



110 



MODELS AND SHAPING 



[Chap. 4 



when the dissipated energy is restored in the capacitor, the circuit auto- 
matically adjusts itself so as to avoid operating in the positive grid 
region. The negative grid clamping is reflected, through the controlled 
source, into effective positive plate-circuit clamping as illustrated by the 
waveshapes of Fig. 4-13. Note that operation over the complete grid 
base is guaranteed before the tube cuts off. Of course, a small amount of 
distortion appears as a consequence of the small grid conduction angle, 
but if this can be tolerated, then this circuit represents an extremely 
simple way to self-bias a tube and still ensure the maximum possible 
voltage swing. 



M 



rw 



vw 




Fig. 4-13. Grid-circuit clamping and resultant waveshapes. 

We might further observe that any capacitor-coupled amplifier will 
exhibit a measure of grid-clamping action when the input signal over- 
drives the grid, i.e., when the positive peak of the input is greater than the 
cathode bias voltage. The additional bias component produced shifts 
the operating point away from the positive grid region, and provided that 
the tube is not driven into cutoff, the actual distortion will always be less 
than expected on the basis of a simplified analysis. 

4-3. Triode Gates. Almost any configuration of triodes, coupled 
together by a common load resistor in either their cathode or their 
plate circuit, will serve as an and or an or gate (Figs. 4-14 and 4-16). 
Since the individual tubes are essentially in parallel, the basic considera- 
tion for proper functioning as an or gate is that each input pulse should 
be able to make its presence known at the output. If the tubes are 
normally biased below cutoff, then the conduction of any one tube, 
brought about by an input pulse, will operate the gate and produce a 
change in the circuit state. 

On the other hand, in an and gate the effects of a single excitation 
pulse must be swamped out by the normal conditions of the other 
parallel tubes. In biasing them in their saturation regions the aggregate 
effect of the load current contributed by all the tubes would not be 
seriously disturbed if, upon the injection of a negative pulse, a single tube 



Sec. 4-3] simple teiode and teansistoe cieguits 111 

switches from saturation to cutoff. The major change of circuit state 
will occur only when all the tubes are driven off. 

Triode OR Gate. We now propose to examine the operation of the 
common-cathode or gate of Fig. 4-14o. Besides calculating the steady- 
state response from the piecewise-linear model, we are also interested 
in finding the rise and decay time constants of the output waveshapes. 
All tubes are normally maintained completely cut off by returning 
the grid resistors to a large negative bias (E cc <K 0). The input pulse 
is applied through the R\C\ time constant, which may be assumed long 




Fig. 4-14. (a) Common-cathode triode ob gate; (6) piecewise-linear model of a single 
on tube. 

enough, with respect to the duration of the driving signal, so as not to 
introduce any additional shaping. We shall further assume, purely for 
the ease of calculation, that the single pulse applied to any grid will drive 
that tube to the verge of saturation. 

The grid-to-ground voltage corresponding to saturation may be found 
by setting e e = in the piecewise-linear model of Fig. 4-14b. Solving 
the reduced network for the equivalent input and output voltages leads to 



E a = Eht — 



Rl 



r p + Rl 



Ehk 



(4-16) 



However, the grid was initially at E cc , and since we wish to raise it to 
the value given in Eq. (4-16), the input pulse height would have to be 



B* 



Ei,, — Ea 



112 



MODELS AND SHAPING 



[Chap. 4 



The output due to smaller or larger inputs could be calculated, quite 
easily, once values are assigned to the circuit parameters. 

The output rises toward this steady-state value with a time constant 
determined by the output load and the internal impedance of the' tube. 
Since, in the circuit configuration shown, the controlled source includes 
the output voltage as one of its components, the value of the output 
impedance is not immediately obvious upon inspection of the model. 
By proceeding from the basic definition, i.e., the output impedance is 
equal to the ratio of open-circuit terminal voltage and the short-circuit 
current, its evaluation is straightforward. Furthermore, the d-c sources 
may be ignored since they only set the quiescent operating point. 

By removing the external load Rl || C„ the open-circuit output voltage 
of Fig. 4-14b is given by 



e 



tie c = m(«i — e ) 



(4-17) 



In finding the short-circuit current, we return the cathode directly to 
ground, thus removing the dependency of the controlled source upon any 

voltage appearing in the output loop. 




Therefore 






(4-18) 



Solving Eq. (4-17) for e„ and dividing 
the result by Eq. (4-18) yields 



z ; 



1 



M+ 1 



g m 



(4-19) 



Fig. 4-15. Input and output pulse 
appearing in the common-cathode or 
gate of Fig. 4-14a. 



ance, on the order of 400 to 
constant is quite small: 

Tl 



700 



The above approximation holds for a 
high-/* tube (i.e., where m + 1 = /*)• 
The single tube considered to be in 
its active region is operating as a 
cathode follower, and Eq. (4-19) ex- 
presses its very small output imped- 
ohms. Consequently, the rise time 



( Bl|l iTTi) 



c, 



where C, represents the total stray capacity appearing across the output. 
Upon removal of the excitation, the tube cuts off and the output 
decays back toward zero with the new, much longer time constant 



Tj = RlC, 



Sec. 4-3] 



SIMPLE TRIODE AND TBANSISTOE CIRCUITS 



113 



which is simply that of the external load. The input and output wave- 
shapes sketched in Fig. 4-15 illustrate the pulse distortion introduced 
by this gate. Usually Rl is kept small so that the trailing edge will 
not be smeared excessively and therefore falsely indicate the presence 
of a gating signal after its removal. 

The particular gate of Fig. 4-14 was designed for positive pulse excita- 
tion; for gating negative pulses we would turn to a common plate con- 
nection and apply the negative input pulses in the cathodes of the 
individual tubes. Under these circumstances, the triode would amplify 
the input pulse, but with its larger output impedance it would also 
introduce a greater degree of distortion in the initial rise of the output 
pulse. 




iE cc (E cc >0) 
Fig. 4-16. Common plate connection — and gate. 

Triode and Gate. An alternative mode of gate operation is typified 
by the and gate of Fig. 4-16. The grids, returned to the positive bias 
Ea, through R lt are normally maintained in their saturation region, with, 
however, their self-clipping action limiting the maximum positive grid 
excursion to 



E„ = 



E a 



r e 



Ecc^O 



(4-20) 



Ri + »\s " Ri 
where Ri » r . 

Upon the excitation of any one grid, by a negative pulse, that par- 
ticular tube will cease conduction, and since its plate current no longer 
flows through R L , the composite plate voltage rises slightly. Provided, 
however, that R L » r„ this change in voltage will be almost negligible. 
Only after every tube is simultaneously cut off does the plate voltage 
change by an appreciable amount. It rises from its initially low satura- 
tion value of 



Eu^ 



Rl + r„/n 



Em 



(4-21) 



which is found from the piecewise-linear model holding when n tubes are 
saturated — to the final value of E». 



114 



MODELS AND SHAPING 



[Chap. 4 



One obvious peculiarity of this and gate is that it produces a positive 
output upon excitation by negative input pulses, and if we wish to drive 
a second, similar gate, a buffer amplifier must be inserted to invert the 
pulse. If the circuit is modified so that positive driving pulses are 
injected into each cathode, the gate output will be an amplified pulse 
of the same polarity as the input. 

Amplitude Gates. Triode amplitude gating is predicated on the con- 
trol signal (normally injected at the cathode) shifting the operating 
point of the cutoff tube into its active region. At this time the amplifica- 
tion from grid to plate permits the transmittal of the information signal. 
The basic gating behavior is illustrated by the simplified circuit of Fig. 
4-17a. But to avoid loading the control-signal generator by the low out- 
put impedance seen in the cathode circuit, the control signal might well be 




°E K 

(a) (6) 

Fig. 4-17. (a) Basic amplitude gate; (6) cathode-coupled controlled gate. 

coupled through a cathode follower. The resultant circuit (Fig. 4-176) 
is of the identical configuration with the cathode-coupled clipper of 
Fig. 4-11, with, however, signals applied to both grids. 

Under normal conditions, the positive d-c level of the control signal 
raises the common-cathode voltage to a value that is high enough to 
ensure biasing Tt well below cutoff. For optimum performance, the 
response of this gate should be independent of the control-signal ampli- 
tude; to this end, the gating signal simply drives Ti into its cutoff region, 
allowing T% to function as a normal amplifier. If the above conditions 
are satisfied, the calculation of the output voltage, with respect to the 
gated signal e-„ may be carried out without reference to TY 

The control pulse still makes its presence felt at the output, although 
somewhat indirectly. As seen in the waveshapes of Fig. 4-18, the 
abrupt change of state of T2, which takes place when the tube enters 
into its active region, effectively superimposes the gated output on a 



t 



Sec. 4-3] simple triode and transistor circuits 115 

pedestal having the same duration as the gating pulse. Under many 

circumstances such an additional term would be intolerable. It follows 

that in order to eliminate the output pedestal, we must maintain the 

current flow through the output load invariant with respect to the state 

of T 2 . We accomplish this by 

adding a third triode (T 3 of Fig. 

4-19) which is gated off at the same 

time Ti is gated on. Since T% 

normally conducts and since it 

time-shares the output load, we 

can adjust its plate current, by 

varying JB«, so that the output 

voltage remains constant across 

the change of circuit state. The 

application of a negative gating 

pulse drives both T\ and T x into 

cutoff. These tubes are effectively 



(a) 



Ekb 

t 

eo 



M 



m/wT 



AAA/WWWWWWVWl t - 
(b) 

Fig. 4-18. Cathode-coupled gate wave- 
shapes, (o) Control signal; (6) input 
signal and gate output. 



removed from the circuit, and the transmission path, now established 
through Ti, is unaffected by any other triode. As the waveshapes testify, 
the output voltage no longer changes abruptly; the pedestal is now 
missing. 



[ftp 



# 



£ 




t 



-A/V- 



: 3 4 |\AA/Vr 




Fig. 4-19. Circuit for elimination of the output pedestal. 



In concluding this summary of triode gates, we might note that the 
controlled source, which is inherent within the triode, affords a much 
greater degree of isolation between the input and the output than was 
possible with diode gates. Moreover, this isolation greatly reduces or 
eliminates completely any interaction between the various driving signal 
sources. 



116 MODELS AND SHAPING [CHAP. 4 

4-4. Transistor Models. An n-p-n transistor consists of two junc- 
tions of n material, either silicon or germanium, to which n-type impuri- 
ties have been added. These are formed on each side of the thin layer 
of p material constituting the base of the transistor. One junction is 
normally forward-biased with respect to the base structure, in this case 
by making its polarity negative (v<* < 0), and it serves as the emitter. 
We might observe that structurally the emitter-base junction forms a 
forward-conducting diode. The remaining junction becomes the col- 
lector, and for proper transistor operation, it should be back-biased (posi- 
tive with respect to the base). Furthermore, the collector junction 
current is approximately equal in magnitude to the emitter current (Fig. 
4-20c), but its voltage drop will be much larger. Since most of the 
transistor power is dissipated at the collector, its junction must be com- 
mensurate in size. However, special transistors have been designed for 
bidirectional transmission, and in these the particular junction which 
would function as the emitter would be determined solely by the voltages 
present in the circuit. At any instant whichever junction happens to 
be forward-biased becomes the emitter, and if the other is back-biased, 
it becomes the collector. 

A transistor's characteristics are determined by the particular material 
used, i.e., germanium or silicon, and by the nature of the junction formed. 
The major differences in the volt-ampere curves appear at low collector 
voltages. These variations are relatively minor, and therefore the models 
which shall be derived below are adequate representations of all types of 
transistors. 

Figure 4-20a, the schematic representation of the n-p-n transistor, is 
presented in order to define the circuit polarities which we shall employ 
in the remainder of the text. The ready availability of complementary 
units, i.e., both n-p-n and p-n-p transistors, creates some confusion when 
a current direction is arbitrarily assigned. In an effort to avoid any 
ambiguity, we shall choose the emitter arrow as the reference and assume 
all current flows in the direction it points. If the emitter current flows 
out (as shown in Fig. 4-20a for the n~p-n unit), then the base and collector 
currents flow into the device. Alternatively, in the p-n-p transistor the 
emitter arrow is pointed in the opposite direction, into the transistor, and 
all voltages and currents would be reversed. Thus the transistor may 
be considered as a simple node where 

i. = h + i. (4-22) 

Before attempting construction of a piecewise-linear model for the 
transistor, we must satisfy ourselves as to the suitability of such a repre- 
sentation. Following much the same procedure used in deriving the 
triode model, our starting point will be the examination of the empirically 



Sec. 4-4] 



SIMPLE TRIODE AND TRANBISTOB CIECUITS 



117 



determined transistor characteristics, which are either furnished by the 
manufacturer or obtained by measurement in the laboratory. Even 
though gross approximations may be acceptable in replacing the actual 
characteristics by a family of straight lines, the linearized curves should 



=- U €* 




°cb ~~. 



(a) 



10 v lv 




-0.1 -0.2 -0.3 
Emitter voltage va 
(b) 



8 












, 1 

I e —8 ma 










































1 
• 6 














6 














■*-■ 

c 




























fc 

" 4 














4 














i 




























8 
2 














2 













































If 


flco 


























-2 



20 



24 



8 12 16 

Collector voltage va, volts 
(c) 
Fio. 4r-20. Typical common-base characteristics — junction nrjm transistor, (a) 
Schematic showing current flow and polarity; (6) input volt-ampere characteristic; 
(c) collector characteristics. 

still lead to results consistent with the transistor's physical behavior; 
that is to say, the actual curves must be readily representable by straight- 
line segments and the equal increments of the controlling variable should 
produce roughly equal changes in the controlled term. 

The most superficial glance at the volt-ampere characteristics of the 
typical rir-p^n junction transistor (Fig. 4-20) convinces us that the device 



118 



MODELS AND SHAPING 



[Chap. 4 



itself behaves in an almost piecewise-linear manner. Therefore we 
expect that the model representation will bear a closer correlation to the 
actual characteristics than it was possible to achieve with the triode. 
On the other hand, since the difficulties encountered in the manufacture 
of transistors manifest themselves in a wider divergence of characteris- 
tics, their actual response in a circuit would probably differ, by a com- 
parable amount, from the ideal. By again directing our attention to 
Fig. 4-20, we would draw the following additional conclusions. First, 
the transistor exhibits current-controlled behavior in contrast to the 
voltage-controlled response of the triode. Secondly, we see that the 
collector curves are almost constant-current lines, with the particular 
operating path determined by the emitter current. Finally, from a 



10 - 




. 


10 




* — 8 




6 


1 




tfAi',, 


... -2 


t 

i i i 1 


1 h-P 



5 10 15 20 

V c b, volts 

(b) 

Fig. 4-21. (a) Linearized transistor input characteristic; (6) piecewise-linear collector 
characteristics. 



-V„-0.1 -0.2 

Veb, VOltS 

(a) 



macroscopic viewpoint, the effect of the collector current on the input 
response is not very pronounced; for the initial discussion the feedback 
from the output to the input may be ignored. 

Common-base Model. In finally replacing the actual characteristics 
by the linear segments needed for the large-signal model, we shall approxi- 
mate the set of input volt-ampere curves (Fig. 4-206) by the single line of 
Fig. 4-2 la. Its equation is 



»e6 = -(V + »>n) U > 



(4-23) 



Equation (4-23) corresponds to a biased diode having a forward resistance 
of rn. Often this term is referred to as the large-signal input resistance 
(grounded-base connection). 

The collector characteristics are replaced by a family of parallel 
straight lines having a spacing determined by the emitter current. Their 



Sec. 4-4] 



SIMPLE TRIODE AND TRANSISTOR CIRCUITS 



119 



parametric equation, which we write in terms of the variable of interest, 
becomes 

i, = f(v A ,i e ) = gja + ai, i e > and iu > (4-24) 

The above defining equation (4-24), representing the response of the 
output circuit, is analogous to that written for the triode plate circuit 
[Eq. (4-2)]. The roles of current and voltage are simply interchanged, 
and we shall see that this leads to the dual-model representation. With 
respect to the original characteristics of Fig. 4-20c, the linear slope is 
g c (the open-circuit collector admittance) and the current intercept is 
ai,. Moreover, the forward-current amplification factor 



At, J 

At« Iam-o 



(4-25) 



fulfills a similar role to that performed by n in the vacuum tube. Both 
coefficients may be found by measuring the collector current, first, 
with an applied collector voltage and the emitter open-circuited (g c ), 
and second, with an applied emitter current and the collector short' 
circuited (a). 



"eb 






o ci 



Fig. 4-22. Fiecewise-linear model — n-p-n junction transistor. 

We can construct the piecewise-linear model shown in Fig. 4-22 from 
the above defining equations. The forward-current transmission appears 
in the output circuit as the current generator ai,. Diodes are inserted 
to restrict operation to the appropriate regions as well as to represent 
the transistor's behavior under saturation and cutoff conditions. We 
consider that the transistor is cut off when the input diode is back-biased 
and that it is saturated once, the collector diode conducts. These con- 
ditions lead to the following limits for the active region: 



v* < — Ve Vcb>0 



(4-26) 



When the external emitter supply voltage is relatively large, the 
error in subsequent calculations due to omitting the small constant 
emitter drop will be insignificant. Moreover, the treatment of the input 
circuit as a constant resistance will greatly simplify any analysis. Except 



120 MODELS AND SHAPING [CHAP. 4 

when using low supply voltages, the small drop will be ignored and the 
emitter will be assumed to conduct at zero volts. The values of the 
parameters of the junction transistor normally he within the following 
bounds: 

r u = Ha ^ 10 to 100 ohms 

Vo = 0.05 to 0.4 volt 

ffc = Ha, = - ^ 10-* to 10- 7 mho 

a = h fh £* 0.95 to 0.99 

The h parameters tabulated above are those normally given by the 
manufacturer; the first subscript indicates the terminal under considera- 
tion (i for the input and o for the output), and the second subscript, the 
common grounded terminal (b for the base and e for the emitter). 

If we refer back to the collector characteristics, we shall observe 
some small collector-base current flow even when i t = 0. This term 
is ho, a temperature-dependent current which is due to the minority 
charge carriers present in the transistor material and which might be 
equated to the reverse current flow of a semiconductor diode. At room 
temperatures I e o would be only a few microamperes, but since it roughly 
doubles for every 10°C rise in the ambient temperature, it would become 
important at elevated temperatures. Its equivalent-circuit representa- 
tion would be an additional current generator in parallel with the con- 
trolled source shown. In the interests of simplicity, I c o will be omitted 
from the following discussion; the reader can always include this term 
where necessary. 

Common-emitter Model. Many transistor circuits exist where the input 
current is injected into the base and where the emitter replaces the base 
as the reference terminal. Consequently, it would be of interest to 
construct an alternative large-signal model in which the base current is 
the controlling term. 

This transformation follows directly from the model of Fig. 4-22. We 
might first note that 

4 = i. - ic = (1 - <*)i. (4-27) 

Solving Eq. (4-27) for *, and multiplying the result by a yields 

cd. = r -^— i b = fa (4-28) 

1 — a 

where /S is identified as the base-to-collector current-amplification factor. 
Since a is close to unity, taking on the values given above, normally 
ranges between 20 and 100. 

To complete the transformation from the emitter-controlled circuit 
of Fig. 4-22 to the base-controlled model of Fig. 4-23, we must ensure 



Sec. 4-4] simple tkiode and tranbistok circuits 121 

that all the remaining terminal characteristics of the two models are 
identical. In both, the emitter-to-base voltage drop must be the same 
for a given emitter current. For the grounded-base circuit, 

v * = — (Vo + iSu) 

In the grounded-emitter configuration 

vt. = -»* = +Fo + w-it (4-29) 

Substituting the value of t» given in Eq. (4-27) into Eq. (4-29), 

vt. = Vo + (1 — «Kit. 

Thus, by comparing appropriate terms, 

_/ _ *"u 



1 - a 



- (l + Mm 



(4-30) 



Under saturation conditions (conduction of the collector diode) the 
controlled source is shorted, is removed from the circuit, and r' n reduces 
to rji. 






ri'i-fl+UMi 

w\ 



rOi 



"V 



V- 



-VW- 

-w- 



»a 



Fig. 4-23. Large-signal transistor model — common-emitter equivalent circuit. 

The single remaining unknown term in the new model is the relative 
value of the output conductance tt. Once we set i, = 0, in the circuit 
of Fig. 4-22, we can then write the simple equation 



»<* = tefc 



h. 



(4-31) 



where the small constant term To is neglected. From Fig. 4-23, under 
the same conditions of an open-circuited emitter, 



lb = — tc 



But the current source in the output loop is controlled by the base cur- 
rent flow, which in this case is —i c . Substitution into the model of 



122 



MODELS AND SHAPING 



[Chap. 4 



Fig. 4-23 results in the following output equation: 

Vce = t«(l + /3)rd 



(4-32) 



Since the emitter-to-base voltage is extremely small, «<* = t>„. By equat- 
ing Eq. (4-31) to (4-32), we can find the internal impedance of the base- 
controlled current source. It becomes 

r. 



rj = (1 — a)r c = 



+ 1 



(4-33) 



and rd is only one-tenth to one one-hundredth part of r c . 

The reverse collector current / c o also flows through the base circuit 
and with no input 

ib — ~ Ico 

This term will also be multiplied by the base amplification factor 0. 
For consistency of current flow, the controlled source must be shunted 
by an equivalent current generator of (1 + p)I c o- The parallel combina- 
tion of these two generators adds up to the actual current flow Z c o. 
10 































£g 


OfiA 






















V76 
























\ 


50_ 
























\'t 


6_ 
























_10C 


_ 




















75. 


























j0_ 
























1 


25 























■ 




1 







3 



16 



20 



24 



12 

V C e, VOltS 
Fig. 4-24. Transistor-collector volt-ampere characteristics — common-emitter con- 
figuration. 

The nonlinear attributes of the model must remain invariant under 
the circuit transformation, and therefore the two diodes remain in the 
emitter and in the collector arms. This is consistent with physical 
behavior since the emitter-base and the collector-base circuit each super- 
ficially constitute a diode, one forward-biased and the other back-biased. 

If we started the construction of the transistor models from the volt- 
ampere characteristics which are measured for the common-emitter 
configuration (Fig. 4-24), the model resulting would be identical with 



Sec. 4-4] 



SIMPLE TRIODE AND TRANSISTOR CIRCUITS 



123 



Fig. 4-23. We note that the slope of the collector curve is much steeper 
than that of the common-base circuit, thus verifying the result of Eq. 
(4-33). Moreover, as we anticipated in Eq. (4-28), the change in collector 
current with respect to changes in base current is much larger than with 
respect to the former controlling changes in the emitter current. 

The comparative h parameters for the terms in the model of Fig. 4-23, 
together with their range of values, are 

r' u = H ie = 100 to 1,000 ohms 
1 



9d = 



Td 



Ho, = 2 to 100 jtmhos 




= H„ = 20 to 100 

Incremental Models. When dealing with small signals it is occasionally 
necessary to account for the effects on the input circuit of the changing 
collector voltage. In the large-signal 
model this factor was neglected, lead- 
ing to the simple input circuit of Fig. 
4-22. To take it into account in the 
incremental model, we can insert a 
voltage-controlled source as shown in 
Fig. 4-25. The reverse-voltage ampli- 
fication h^, is very small, on the order 
of 10 -6 to 10~ 8 . For a more accurate 
representation, the other parameters (a, r e , and rn) may be reevaluated 
at the operating point. 

For the model of Fig. 4-25, the two defining equations are 

t><* = — riii, + have (4-34o) 

ic = od e + g<Pd> (4-34b) 

where the voltage and current terms represent incremental variations 
about the operating point. From Eq. (4-34a) we see that the small 
reverse-voltage transmission is defined as 



Fig. 4-25. Hybrid transistor model — 
common-base connection. 



hrt, = 



AVeb 



(4-35) 



Inspection of Fig. 4-206 would indicate that h T b increases with decreasing 
collector voltage and decreases with decreasing emitter current. In any 
calculations, it must be evaluated at the known operating point. 

If the two-source hybrid model of Fig. 4-25 is replaced by the equivalent 
T network of Fig. 4-26, one of the controlled sources will be eliminated. 
Yet its effect is still present, represented instead by the common-base 



124 MODELS AND SHAPING [CHAP. 4 

resistor »v The equivalents of the two models are found by solving the 
terminal response under identical operating conditions. 

For a short-circuited output, the input equation of the hybrid model 
[Eq. (4-34a)] reduces to 

f* = — r n i. 

The equivalent equation found from Fig. 4-26 is 

»* = -[r.+ (1 -«)*]*. (4-36) 

where r' c is very large compared with r b and may be neglected. Thus 

r, + (1 — a)r h = r u 
and for a very close to unity, r, very closely approximates ru. 





Fig. 4-26. Incremental T model — com- Fio. 4-27. Incremental T model — com- 
mon-base connection. mon-emitter connection. 

By solving for Vj, from the models of Figs. 4-25 and 4-26 at i. = 0, 

h <» = Hr? = p (4-37) 

r h + r c r e 

Following the same procedure with respect to the collector-to-base 
terminal response and remembering that r' c » r b , the remaining equiv- 
alents are 

r' c S r c and a' = a (4-38) 

The values of the two new parameters generally lie within the range 

r b , from 100 to 400 ohms 
r„ from 10 to 50 ohms 

and in much the same manner as htb, n varies widely with emitter current 
and collector voltage. It may decrease by a factor of 2 to 10 as we 
drive the transistor from close to cutoff to close to saturation. 

The common-emitter T model (Fig. 4-27) will be almost identical 
with the common-collector model, with the major difference appearing 
in the collector branch. As in the equivalent hybrid model, r<j replaces 



Sec. 4-5] simple triode and transistor circuits 125 

r c and /3 replaces a. Furthermore 

r' n £* r h + (1 + /S)r. 

Large-signal T Models. As an alternative to developing the incre- 
mental model from the large-signal representation, we can begin by 
examining a small section of the volt-ampere plane and, as we expand 
the area under review, modify the incremental circuit toward a large- 
signal model. Our starting point might be one of the T circuits of 
Fig. 4-26 or 4-27, and two limiting diodes would be added to restrict 
the permissible operating region. If necessary, a bias battery Vo can 
also be inserted in the emitter circuit. Since a model is, at best, a reason- 
able approximation of the actual response, if average values are chosen 
for the T parameters, the circuits of Fig. 4-28 will be as serviceable as the 
hybrid models of Figs. 4-22 and 4-23. 



o — |4_AAA 





(") (b) 

Fig. 4-28. Piecewise-linoar T models, (a) Common-base connection; (6) common- 
emitter connection (Fo and I c o terms omitted). 

4-5. Simple Transistor Circuits. The transistor, even more so than 
the triode, is readily adaptable for clamping and clipping applications. 
Since structurally it may be likened to a pair of diodes back to back, we 
might use either one independently in any of the circuits of Chap. 2. 
As a single diode is more economical, this is at best a dubious choice. 
But if advantage is taken of the current source shunting the collector 
diode to establish the location of the clamping level or the clipping point, 
as in Fig. 2-19, then the necessity of introducing an external bias gen- 
erator will be avoided. Furthermore, since complementary transistors 
exist, n-p-n and p-n-p, we are no longer restricted, as we were with the 
triode, to unidirectional operation. 

It is of interest to calculate the over-all transfer characteristics and, 
in the process, to delineate the various regions of transistor behavior. 
At the same time an attempt will be made to simplify the two-loop models 
of Figs. 4-22, 4-23, and 4-28 to the point where they become almost 
ridiculously simple. Of course, the approximations made must be kept 



126 MODELS AND SHAPING [CHAP. 4 

in mind because many transistor circuits may not lend themselves to this 
treatment. The many others that do, including most of the circuits 
which will be treated in later chapters, justify our spending some little 
time on this topic. 




Fig. 4-29. (a) Transistor amplifier circuit; (6) piecewise-linear model. 

Consider, for example, the transistor amplifier circuit of Fig. 4-29o. 
In drawing the T piecewise-linear model (Fig. 4-296), we also converted 
the circuit, external to the base, to its Norton equivalent, where 



Ri = R, || Ra 



and 



I _ Eg 
Ra 



Before beginning the analysis, we might note that the percentage of the 
driving current which actually flows into the base depends on the relative 
magnitudes of the transistor base input impedance Z\, and the external 
base load JRi. 

R, 



Ri + Zb 



ii{t) 



(4-39) 



It thus seems logical to evaluate the over-all transmission in two steps: 
first, the direct transmission from the base terminal to the output and, 
second, the transmission factor relating the driving source and the base 
current. The over-all forward transmission is the product of these two 
terms. Moreover, since the bias current h only shifts the transfer char- 
acteristics with respect to the zero point, it might be just as easy initially 
to ignore h and, after the solution is complete, then consider the shift in 
axis introduced by this constant-current term. 



Sec. 4-5] 



SIMPLE TRIODE AND TRANSISTOR CIRCUITS 



127 



The first operating region with which we shall be concerned is the cut- 
off zone, characterized by the back-biasing of the emitter diode. Figure 
4-30o presents the reduced model holding in this region. The input- 
circuit defining equation will enable us to locate the boundary. 



v b . = ibH + OS + 1)4^ + Riib + #i* 
= ibln + OS + l)r„ + R2] + Etb 



(4-40) 



But the reader may verify from Eq. (4-33) that (/3 + l)r d = r c . It now 
seems reasonable to neglect the small voltage drop appearing across r b 




Fig. 4-30. (a) Model defining cutoff region; (6) approximate representation. 

while considering the bounds of this region (r c ^>r b ). For the emitter 
to be back-biased v be < 0; solving Eq. (4-40) establishes the cutoff region, 
in terms of 4, as 

' Evb ~" (4-41) 



4< 



r.+ ffi 



= 



The current flowing in the base under cutoff conditions is simply the 
reverse current through the back-biased collector-to-base diode. 

The input impedance of the transistor is the coefficient of i b in Eq. 
(4-40). Since the predominant term is r c , which is usually very much 
larger than any external load appearing at either the collector or the base, 
practically all the signal source current will be shunted through Ri. 
Transmission from the input to the output is completely negligible, and 
for simplicity, this path may be replaced by an open circuit. The model 
of Fig. 4-306 results, and we have seen that it represents, quite ade- 
quately, the transistor's behavior under cutoff conditions. 

Next, in considering the active region of transistor operation (Fig. 
4-3 la), we note that for maximum load current, the external load should 
be very small compared with the internal impedance, that is, R 2 « r d . 
Under this condition the current through R 2 will be $i b . Along with 
neglecting the current flow through r A , the resistance itself will be removed 



128 



MODELS AND SHAPING 



[Chap. 4 



from the equivalent circuit. A current of t 6 (l + 0) flows through r„ 
and the input equation for this region becomes 

vu = ibn + (1 + /3)r«4 

= 4[r„ + (1 + /3)rJ (4-42) 

The interpretation given to Eq. (4-42) is that any resistance in series 




6t 



►^l V 



il(t) 




(b)' 
Fio. 4-31. (o) Model defining active region; (b) approximate representation. 




Fig. 4-32. (a) Model defining saturation region; (6) approximate representation. 

with the emitter is multiplied by (1 + 0) as it is reflected into the base 
input circuit. From Eq. (4-39), 

Ri 



and 



% r b + (1 + $)Y. ;+ i?i 
ic = /34 = 



»'i(0 



H + (1 + |8)r. + «! 



»'i(0 



(4-43) 



For the maximum possible current gain, i?i 55> rt + (1 + 0)»V Upon 
satisfaction of this necessary inequality, both the base and emitter 
resistors are replaced by short circuits and the trivial model of Fig. 4-316 
results. 

The upper bound of the active region occurs when the collector voltage 
drops to zero. From the simplified model of Fig. 4-316, 

v„ = En - pibR 2 > 



Sec. 4-5] simple teiode and transistor cibcuits 

Thus the limits of the active region are 

En, 



< % < 



fiRi 



129 



(444) 



As t'x(t) continues to increase, the circuit enters upon its saturation 
region. The collector diode con- 
ducts, and, using the approxi- 
mations previously made, the model 
of Fig. 4-32a may be reduced to 
the simple representation of Fig. 
4-326. We might observe that 
since there is no longer any trans- 
mission from the input to the 
output, the slope of the transfer 
characteristics will be zero. 

Figure 4-33 shows the input- 
output transfer characteristics. In addition, the shift in axis produced 
by the bias current h q is also accounted for by the second ordinate 
(dashed line). 



&n 








Rz 


• X 






I 


/ i 









'w fit 


h(t) 



Fig. 4-33. Transfer characteristics of the 
circuit of Fig. 4-29. 



+20 v 




+20 w 







\ T 1 

i V 

1 1 — 






I 
i 

-r 



a t. 



(d) 



Fig. 4-34. Circuit, models, and waveshapes for Example 4-2. 

Example 4-2. The model techniques are also quite useful in evaluating the time 
response of active circuits containing energy-storage elements. To illustrate the 
methods employed, we shall solve the simple circuit of Fig. 4-34a. The switch, which 
is initially closed, will be opened at < — and closed after steady state is reached. 



130 MODELS AND SHAPING [CHAP. 4 

Before the switch is opened the large base current flow saturates the transistor. 
This statement can be verified by assuming that it is true and calculating the current 
flow. The model of interest reduces to that shown in Fig. 4-346. By taking the 
Thevenin equivalent of the circuit seen looking into the transistor, 

ii(0_) = 2^0 = 50ma 
From the active-region model of Fig. 4-34c, the conditions for saturation are given by 

fin. X 2 K + (0 + 1)4. X 2 K = 20 volts 
or it, = 100 ii& 

and this current is greatly exceeded. 

Solving the circuit of Fig. 4-346, the voltage at the emitter is 

"• (0_) = 2K+VkU2K 20 " 15volts 

After the switch is opened, the emitter starts decaying toward 10 volts, with the time 
constant 

t,-MX2K = 2 msec 

But the base current also decreases as C charges. When it falls to 100 /ia or when 
»„ — 10.1 volts, the transistor becomes active. The elapsed time will be found 
from the exponential charging equation 

v. - 10 + Se-'"i 

by substituting in the final value. Virtually the complete exponential is used, and 
thus 

h = 4n = 8 msec 

After the circuit becomes active, the increase in input impedance to (fi + 1)2 K 
[from Eq. (4-42)] increases the charging time constant to 

t 2 = 1 /if(51 X 2 K + 1 K) = 103 msec 

The discharge continues toward zero with this very long time constant. Finally 
the switch is closed and the transistor switches back into saturation. 

The waveshapes of the base current and the emitter voltage are shown in Fig. 4-34d. 

4-6. Transistor Gates. If we reexamine the basic premise from which 
we constructed the various circuit configurations used as gates, i.e., the 
opening or closing of a transmission path upon signal excitation, it would 
seem that perhaps a series arrangement of active elements would serve 
as well as the parallel gates of Sec. 4-3. With triodes, the high plate 
supply voltage necessary for their proper operation would have made 
any such discussion academic. However, at this point we have at our 
disposal the transistor, an active element requiring very low operating 
voltages, and we may therefore contemplate this alternative configura- 
tion. Of course this does not prevent the use of transistor parallel gates. 

The series gate fulfills its function as the individual transistors switch 
from the almost direct short they present under saturation to an open 
circuit when cut off. Since the opening of a series transmission path any- 



Sec. 4-6] 



SIMPLE TBIODE AND TRANSISTOR CIRCUITS 



131 



where along its length is equally effective, for an ok gate (Fig. 4-35) the 
individual transistors would normally be biased full-on. As any one 
changes state into cutoff, the output path opens and the gate indicates 
the presence of the input pulse. Conversely, in an and circuit (Fig. 4-36), 
all the transistors would be biased below cutoff, with the output terminal 
remaining above ground potential until the simultaneous excitation at all 




e 3 o-VW 



t° 

t° 

e 2 

t° 

Ebb 
t 





Fig. 4-35. Series oe gate and waveshapes. 



e,o-J\/vv 






E bb 






6 Ecc ^ 

Fig. 4-36. Series and gate and waveshapes. 

inputs closes the gate. An important contrast between these two gates 
is that the or gate, being biased on, requires heavy saturation current 
flow under stand-by conditions, whereas the series and gate only con- 
ducts when its logic situation is satisfied. 

One of the major problems facing the designer of logical systems, such 
as digital computers and control equipment, is the power requirements 
imposed by the large number of active elements employed. Besides 
the expense involved in supplying the current drain at the regulated 
voltages needed, the heat produced by the dissipated power must be 
conducted away from the equipment. Otherwise the ambient tempera- 



132 



MODELS AND SHAPING 



[Chap. 4 



ture of the enclosed components may rise to a point where some elements 
will be seriously damaged or even destroyed. To this end, the engineer 
seeks circuits operating with little or no stand-by power. Furthermore, 
since a single triode requires approximately 1 watt of filament heating 
power, transistors have almost completely superseded tubes in many 
critical applications. 

The two transistor gates which best satisfy the above specifications 
are the parallel-connected ok gate and the series-connected and gate. 
Each is normally biased below cutoff. Combinations of these two find 
wide application in the development of systems of binary logic employed 



B 



fcu 




k^T^ 



Fig. 4-37. Direct-coupled complementary transistor gate producing an output upon 
satisfaction of A or B and C. 

in digitally operated equipment. Moreover, complementary design, i.e., 
use of alternating p-n-p and n-p-n gates, permits the output of one gate 
to be directly coupled as the input to the next gate. 

Consider, for example, the circuit of Fig. 4-37, where all the transistors 
are normally cut off: T A , Tb, and T c by having their bases returned 
to the appropriate polarity bias voltage and To by adjusting the network 
of Ri, Rd, and R E so that the voltage from point E to ground will be above 
zero. Satisfaction of the parallel oh gate, upon the positive excitation 
of either A or B, will force the related transistor into conduction, with 
the voltage at point D now dropping to zero. The base of To becomes 
negative, as a consequence of returning the coupling resistor Re to ground 
through T A or Tb, and the series and gate is now primed for conduction. 
As soon as a negative pulse appears at point C the gate opens, allowing 
the output to rise from E ec to zero. If instead a negative output pulse is 



Sec. 4-7] simple tbiode and transistor circuits 133 

desired, the load could be shifted to the emitter-ground circuit of T c , 
with the output now dropping from zero to E cc upon the gate's closure. 

We see that iHhe transistor is allowed to switch between cutoff and 
saturation, then its simplified model representation is an open circuit 
followed by a short circuit. Only in so far as the saturation value of 
base current depends on the transistor parameters will the circuit's 
operation be affected by the particular type of transistor chosen. Pro- 
vided that the threshold value is exceeded for the lowest value of 
expected, the switching performance becomes completely independent 
of the particular active element and would not deteriorate upon the 
substitution of an entirely different transistor. 

4-7. Pentodes. Both the transistor and the triode are three-terminal 
devices, and therefore only two equations, involving four variables, were 
necessary to define their complete operation. Moreover, the approxi- 
mations made in linearizing their characteristics were not so extreme as 
to be incompatible with the actual physical functioning of the device. If 
we attempt to extend this concept to the pentode, we find ourselves in 
some difficulty. We are now forced to consider the related response of 
four terminals instead of two (all voltages are specified with respect to 
the cathode, the reference terminal). Each defining equation will 
involve the terminal characteristics of the remaining three tube elements. 
As a specific example, the equation of plate current must include its own 
voltage as well as terms reflecting the effects of the suppressor, screen, 
and control-grid voltages on the plate current flow. These latter terms 
appear as controlled sources in any model drawn. 

h = /(eiAiAsAs) (4-45) 

Analogous equations could be written for the other pentode elements. 

First of all, in order to expand Eq. (4-45) into a form amenable to 
expression as a piecewise-linear model, we would have to examine the 
curves of plate current and voltage drawn with each of the three grid 
voltages as a parameter. From these families of characteristics we 
might be able to estimate the relative error introduced by linearization. 
After replacing the actual curves by straight lines, the coefficients of the 
linear equation are evaluated from the various slopes and intercepts. 
Since the manufacturer does not usually make such data available, the 
engineer would have to perform his own measurements. As the final step, 
one or more biased diodes would be inserted to restrict the range of 
operation at the plate. 

Assuming that reasonably satisfactory linear operation is possible, Eq. 
(4-45) may be expanded so that it leads to a current-source representation. 

U = ff,A + guei + j/acj + gufit (4-46) 



134 



MODELS AND SHAPING 



[Chap. 4 



The numerical subscripts indicate the particular grid under consideration. 
Each of the coefficients represents the transfer admittance from the 
appropriate grid to the plate. If we restrict operation to the first 

quadrant, the model representation 
of the plate circuit is as shown in 
Fig. 4-38. 

A similar process carried out for 
each element of the pentode will 
result in similar models. In solv- 
ing any specific problem, all four 
models would have to be solved 
simultaneously and the answers 
individually checked to ensure that 




Fio. 4-38. Model representation for the 
plate circuit of a pentode. 



none of the boundary values are exceeded. 

It appears that the desire for generality in this case has greatly increased 
the complexity, even to the point of allowing the model to obscure the 
actual physical processes. Therefore we might best deal directly with the 
characteristics rather than bother to construct a model. 

In many specific circuits, however, the tube voltages are so restricted 
that simpler models may be drawn. The suppressor primarily serves 
to determine the division of the cathode current between the plate and 
the screen grid, while the total 
cathode current depends on the 
screen-grid voltage. If both of 
these are held constant, then 
the plate volt-ampere character- 
istics are almost constant-current 
curves, with the particular value 
of current determined solely by 
the control-grid voltage (Fig. 
4-39). The model for the plate- 
circuit operation would be reduced 
by combining the two constant 
terms of Fig. 4-38, gy,ei and g 3 be 3 , 
into a constant-current generator 
/„ equal to the current intercept 
of the zero grid-voltage line. 

Each of the pentode's grids has sufficient control over the plate current 
flow so that if any one or all are made highly negative, the tube is 
completely cut off. We can thus utilize any individual grid as a gate 
input, and for special gating functions we may employ various combina- 
tions of the three grids. They function in series as valves. Changing 
any one from the on to the off state gives us the or gate. If all elements 



20 ma- 


H 




h 




-£<*- 

2 

. 4 


lOma- 




6 

8 

10 

_ 12 










lOOv 200 300 c » 

Fig. 4-39. Pentode-plate-circuit volt-am- 
pere characteristics — constant screen and 
suppressor voltage. 



Sec. 4-8] 



SIMPLE TBIODE AND TRANSISTOR CIRCUITS 



135 



are normally biased off, they must all be turned on before plate current 
will flow (and gate). Another commonly used pentode circuit is the 
controlled gate of Fig. 4-40. Here the suppressor will maintain the plate 
cutoff until the application of a control pulse. Once current flows, the 
tube operates in a normal manner, amplifying and transmitting the con- 
trol-grid signal to the plate circuit. 

We note that advantage is taken of the control characteristics in much 
the same manner as the series arrangement of transistors. Each control 
element, reflected as a controlled source, has the ability to turn the device 
on and off, and by acting in unison, they create the logical situation 
sought. 

t 




"•"We 



^A 



Fig. 4-40. Pentode-controlled gate. 

4-8. Summary. The importance of the model concept in freeing the 
engineering viewpoint from the conformity imposed by the rigor of 
graphically presented data cannot be overstressed. It is extremely 
difficult to give the imagination free reign when the active element, the 
heart of electronics circuit design, must be treated in a manner different 
from its associated circuit components. Constant referral back and forth 
from the tube or transistor graphics to the circuit equations involving 
the passive elements denies the grasp of the basic nature of the system 
behavior; it leads to the treatment of each circuit as a separate entity. 

In direct contrast to this approach, the very nature of the simpli- 
fying approximations made when drawing a model must lead to a simpler 
presentation of the phenomena under examination. Furthermore, the 
essential unity of circuits and systems immediately becomes clear from 
the similarity of their models. Regardless of the type of control, be it 
voltage, current, pressure, velocity, or temperature, an equivalent con- 
trolled-source representation places the controlling element in the fore- 



136 MODELS AND SHAPING [CHAP. 4 

front. After the piecewise-linear model of the nonlinear element is drawn 
and the remaining circuit components are placed in their proper relation- 
ship, then, within each region, analysis proceeds as in a linear circuit. 
All linear-circuit theory becomes applicable, and many methods that 
would not normally be applied give power to the engineer. 

PROBLEMS 

4-1. (a) Construct a piecewise-linear model to represent the 12AU7 which will 
agree exactly with the tube characteristics at E c = — 10 and h = 15 ma. Give all 
parameter values. 

(6) Superimpose your model on the tube characteristics of Fig. 4-1 and delineate 
the regions where the agreement is within 20 per cent of the actual current. 

(c) Construct an alternative model which will hold for currents less than 5 ma. 
How does it differ from the one found in part a? 

4-2. (a) Drawing on the answers of Prob. 4-1, construct a model which will repre- 
sent the plate characteristics by two contiguous segments. In this manner we can 
obtain closer agreement with the measured curves. 

(6) For this model repeat part 6 of Prob. 4-1. 

4-3. The 12AU7 is used as a simple plate-loaded cathode bias amplifier with Rl = 
12 K and En = 250 volts. Calculate the incremental gain as found from the piece- 
wise model and from the actual characteristics at the following bias voltages: +5, 0, 
—5, —10, and —15 volts. Express the deviation from the value found from the 
model as a percentage. (Assume that r„ = 1,000 ohms in the positive grid region and 
that the signal source impedance is 2,000 ohms.) 

4-4. If we restrict operation to et > ke c (as shown in Fig. 4-2), show that the 
approximate plate-circuit model for e& < ke c is simply a resistor of kr p /(fi + it). 

4-6. Calculate and plot the transfer characteristic for the circuit of Fig. 4-5a when 
the cathode bypass capacitor is removed. The tube parameters are r p = 70 K, 
It ■> 100, and r c = 1 K, and we choose a load resistor of 200 K with E b b ■» 250 volts. 
What value must we specify for R t if the quiescent current is to be 0.5 ma? 

Sketch the output if the input is a triangular wave of 50 volts peak to peak. 




4-6. The three circuits of Fig. 4-41 illustrate the most common amplifiers. The 
same tube and load resistor are used in each case (r„ =■ 25 K, p — 50, Ri = 50 K, and 
r. - 1 K). 



SIMPLE TRIODE AND TBANSI8T0B CIKCUITS 



137 



(a) Draw the complete transfer characteristics of all three circuits on the same axis. 
(6) Specify the output impedance of each amplifier in each of the three regions of 
operation. 

(c) Specify the input impedance of each circuit in each region. 

(d) Can you draw any conclusions as to the applicability of these amplifiers? 
4-7. (a) Calculate and plot e„ versus time for the circuit of Fig. 4-42 when C = 

200 «if and R = 25 K. The input signal is periodic. 

(6) Repeat part a with C - 2,000 ntf and R - 250 K. 

(c) Repeat part 6 when R is connected between the grid of 7/j and 300 volts instead 
of being returned to ground. 



+300 v 



«* 



-70 



*300_ 
/»sec 


200 
/jsec 








t 








Tjand T 2 : 12AU7 



Fig. 4-42 



4-8. At what time after the switch is opened will the tube in Fig. 4-43 start con- 
ducting and how much later will the grid conduct? (p — 20, r T = 10 K, and r, — 
1 K.) Sketch and label the waveshapes appearing at the plate and grid. 



+ 300v 




Fig. 4-43 



4-9. Repeat Prob. 4-8 when the RC combination in the grid circuit is replaced by 
a 10-K resistor and a 100-mh inductance, respectively. 

4-10. In the circuit, of Fig. 4-11, n = 70, r, = 50 K, Eu, - 300 volts, «, - 100 K, 
and Ea " — 150 volts. Find the value of JR. that will ensure symmetrical clipping of 
the input signal. Plot and label the transfer characteristics. 

4-11. Consider the cathode-coupled clipper of Fig. 4-11 having the following 
parameters: 

r, - 70 K ft, - 100 K Eu - 300 volts 



100 



«•- 



JE«. 



-150 volts 



138 



MODELS AND SHAPING 



[Chap. 4 



(o) Calculate the value of R, that wOl result in a quiescent current of 0.6 ma 
through T t . (Hint: Since the common-cathode voltage is approximately zero, the 
drop across R t and T 2 is almost exactly equal to En.) 

(b) Calculate the two clipping levels with respect to the input. 

(c) Sketch and label the transfer characteristic. 

4-12. In the circuit of Fig. 4-41a (Prob. 4-6) a triangular wave is coupled through 
a 0.01-iii capacitor rather than the 10-K resistor. What is the largest possible peak- 
to-peak input signal before the circuit begins to clip the input? Sketch the output 
when the input is twice as large as found above. 

4-13. (a) A cascode amplifier, such as shown in Fig. 4-44, is widely employed in 
television sets where the noise must be kept within bounds. If both of the triodes 
shown are 12AU7's, calculate the quiescent operating point and the transfer character- 
istics. Draw the simplified models holding at each break point. 

(6) Explain the steps which must be taken in order to find the quiescent point by 
a graphical construction. Check the answer to part a by this means. 




Fia. 4-44 



4-14. (a) Calculate the value of .fti that will set the quiescent output of the circuit 
of Fig. 4-45 at zero (E, = E cc = -150 volts). 

(6) What kind of gate is this? What is the minimum input amplitude that will 
produce the maximum output amplitude? 

4-16. In the circuit of Fig. 4-45, Ei is set equal to E hb . The load resistor R t is 50 K, 
and it is shunted by 150 nni stray capacity. 

(a) Sketch the output if we apply — 100-volt 20-msec pulses to ei at t = 0; to e 2 , 
5 msec later; and to e 3 at t = 10 msec. 

(6) Repeat part o if the inputs are changed to 20-volt positive pulses. 

(c) Can you see any way to improve the operation of this gate? 

4-16. Repeat Prob. 4-15 when the load resistor is transferred to the plate circuit 
making the and gate of Fig. 4-16. The cathodes are returned to ground. 

4-17. We wish to eliminate the pedestal present in the controlled gate of Fig. 4-17. 
The tube and circuit parameters are r p = 100 K, p = 100, Rl = 200 K, and JBu, = 
300 volts. The gating tube T 2 is normally maintained cut off by the 12-volt d-c level 
of the gating signal applied at the grid of Ti; superimposed 10-volt negative pulses 
turn Ti on. 



SIMPLE TKIODE AND TRANSISTOR CIRCUITS 



139 



Use a tube with the same parameters and calculate the value of the cathode resistor 
necessary to completely eliminate the pedestal. Assume that the control pulses are 
coupled into its grid through a large capacitor. Specify the amplitude and polarity of 
the signal which must be simultaneously applied to this additional triode. 



?200v 




0,-10 K 

lt-20 

r c -lK 



4-18. There are four possible models which may be used to represent the four 
variables it, tij,,, i c , v„ in the grounded-emitter configuration. Draw these models and 
show where diodes must be inserted to restrict the operation to the proper quadrant. 
Give the meaning of the various circuit parameters in terms of those given in the 
model of Fig. 4-27. 

4-19. The circuit of Fig. 4-46a is used as a current-voltage converter. 

(a) Plot the transfer characteristics (e* versus ij), taking all terms into account, 
and compare it with the plot found from the appropriate approximate models. 

(6) Calculate the input admittance and output impedance in each of the three 
operating regions. 

(c) Plot the output if i'i = 50 sin at ma. 




Fig. 4-46 



4-20. We wish to choose Ri, in the circuit of Fig. 4-466, so as to achieve symmetrical 
clipping. Under these circumstances repeat Prob. 4-19. 

4-21. The transistor used in Fig. 4-46 has an I c „ of 10 ix& at room temperature, and 
it doubles for every 10°C rise. If the value of R\ in Fig. 4-466 is 100 K, by how much 
will the Q point shift with a 20°C rise in ambient temperature? Repeat this calcula- 
tion for the circuit of Fig. 4-46o and express both shifts as a ratio of the Q point 
calculated when I r .o is assumed zero. 



140 



MODELS AND SHAPING 



[Chap. 4 



4-22. The circuit of Fig. 4-47 employs transistors having r. - 30, n - 300, and 
= 20. Sketch to scale the input e t and the output e, when the excitation shown is 
applied. Two cases should be considered: 

(o) When C = 1 /if. 

(6) When C - 0.001 /if. 

9+10 v 



«i 



10 v 



100 
/*sec 




Fio. 4-47 



4-28. Sketch and label the collector voltage in the circuit of Kg. 4-48 if the switch 
is opened at t «■ and closed after the transistor is forced as far into saturation as 
possible. Make all reasonable approximations and indicate the times at which the 
transistor changes state. 

?+20v 




Fio. 4-48 

4-24. As the reverse bias of a transistor drops to zero, the current gain of the tran- 
sistor also decreases. Figure 4-49 shows a semiquantitative view of this decrease for 
the G.E. 2N-123 p-n-p transistor. 

(a) Using Fig. 4-49 and given the data that r, = 28 ohms and n = 80 ohms, find 
the input impedance of the transistor when it is biased such that e„ = 0. 



t bu 

] 40 

e 30 

20 
10, 












s — ■ 


V 




*-§ 







-lv 




-2 


«c 




Fig. 4-49. The transistor parameters are n = 200, r. = 20, r« => 2 megohms, and 
a - 0.95. 



SIMPLE TBIODE AND TRANSISTOR CIRCUITS 



141 



(6) The maximum emitter current ia to be 1 ma. What values of Ri and Rl will 
just meet this requirement? 

(c) What value of ei is necessary to just cut off the transistor? 

4-25. The circuit of Fig. 4-46o is modified by connecting a 1-henry choke from the 
emitter to ground. The input is a 10-kc current sine wave of adjustable amplitude. 
If the signal distortion due to the periodic charge and discharge of the coil is neglected, 
what is the peak input amplitude before the output appears clipped? 

4-26. A parallel combination of transistors is connected as an and gate similar to 
the one shown for the triodes in Fig. 4-16. Their collectors are connected together, 
and the output load resistor inserted in this circuit. We desire to drive this gate with 
negative pulses at the base. 

(o) If we have both n-p-n and p-n-p transistors, a 2,000-ohm load resistor, and two 
20-volt batteries at our disposal, what circuit would be used? 

(6) How large must the input switching pulses be for the maximum output swing? 
The voltage pulses are applied through 20-K resistors directly into the base. Any 
bias resistors selected must furnish a base current of five times the saturation value. 

(c) Repeat this problem if the pulses are injected into the base and the output load 
is connected from the common emitter to ground. Transistor parameters are 



t% = 150 ohms 

a - 0.98 



r, = 30 ohms 
r„ — 2 megohms 



Make all reasonable approximations to simplify your calculations, 

4-27. Using either n-p-n or p-n-p transistors having the parameters given in Prob. 
4-26, we desire to construct a parallel or gate. It should operate on positive signals 
injected at each emitter and should furnish an output across a 2,000-ohm resistor in 
the common-base circuit as well as across a 1,000-ohm resistor in the common-collector 
circuit. 

(o) Draw a three-transistor circuit when two 25-volt batteries are available for 
power. 

(6) If the bias resistors are 2 K, what is the impedance seen by the driving signal 
source? 

(c) The input signal at one emitter is 10 per cent above the value necessary to 
drive the transistor into saturation. Specify this signal and draw both outputs to 
scale. 

4-28. In the gates of Figs. 4-35 and 4-36 the transistors used are those given in 
Prob. 4-26. In both cases Rl - 2 K, R t = Ri = R, = 20 K, E» = 20 volts, and 
Ece - -20 volts. 



+•» 



3v 



&msec 



4v 



i e 2 



2 4 

t, msec 

Fig. 4-50 



«3 



::2i 



OT 15" 

fcmsec 



(a) We apply the signals given in Fig. 4-50 to these inputs through a 2,000-ohm 
source impedance. Sketch to scale the resultant outputs. (Assume that the signal 



142 



MODELS AND SHAPING 



[Chap. 4 



polarity is inverted for the ok gate and make all reasonable approximations in your 
calculations.) 

(6) Calculate the input impedance seen at es, in both gates, under each condition 
of operation. 

4-29. Figure 4-51 shows an n-p-n gate directly coupled to a p-n-p gate. When no 
pulse is applied to the first gate, both gates are "on," that is, v e = 0. When, however, 
a negative pulse appears at the input of T u both gates are cut off. 

(o) What is the approximate input impedance of each transistor? 

(6) When €i = 1.5 volts both transistors are just cut off, and when ei = both 
are just saturated. What values of R lt Ri, R lt and R, are necessary to produce this 
effect? Make all necessary valid assumptions to simplify the analysis. 



?+1.5v 




1.6 w 



r 6 =200 

r e =40 

„ /3foc«OJ-9 

Fig. 4-51 




r„=80 
r e =28 
/Sd^O-lO 



4-30. The circuit of Fig. 4-52 is adjusted so that each transistor is normally biased 
cut off. Sketch the output waveshapes at all three terminals in proper time sequence, 
specifying all voltage values. 



f+4v 
e l 

_4v 



10 K 



10 



5 15 



10 K 



10 K 




Fio. 4-52 



4-31. The two transistors in the gate of Fig. 4-53 are characterized by the two 
curves shown. They hold when the collector is back-biased, but when it is forward- 
biased, we can take v„ = 0. 



SIMPLE TRIODE AND TRANSISTOB CIRCUITS 



143 



(a) If the excitations are as specified, sketch the outputs at e 2 and e> to scale, giving 
all values. 

(6) Repeat part a if the collector supply voltage is changed to —3 volts. 



t 



-300 mv 
+ 600 mv 

t 


-150mv 
-300mv 



12 3 4 5 6 7 
















t, 
msec 



















12 3 4 5 








— L_l 




6 7 i 






msec 











200 



te 



30 
ma 



va, mv 200 300 

Fig. 4-53 



vtt» mv 



4-82. Design a pentode and circuit such that there will be an output if two inputs 
are positive and there will be no output if either of the two inputs or both inputs are 
zero. Draw waveforms showing all these possibilities. Specify the inputs and out- 
puts. Numerical answers are not required, but the relative values should be stated, 
i.e., which voltages are negative and which are positive and whether one voltage should 
be much higher than another for proper operation. 



BIBLIOGRAPHY 

Angelo, E. J., Jr.: "Electronic Circuits," McGraw-Hill Book Company, Inc., New 

York, 1958. 
Clarke, K. K., and M. V. Joyce: "Transistor Circuit Analysis," Addison- Wesley 

Publishing Company, Reading, Mass., in press. 
Ebers, J. J., and J. L. Moll: Large Signal Behavior of Junction Transistor, Proc. IRE, 

vol. 42, no. 12, pp. 1761-1772, 1954. 
Lo, A. W., et al.: "Transistor Electronics," Prentice-Hall, Inc., Englewood Cliffs, 

N.J., 1955. 
Middlebrook, R. D.: "An Introduction to Junction Transistor Theory," John Wiley 

& Sons, Inc., New York, 1957. 
Shea, R. F.: "Transistor Circuit Engineering," John Wiley & Sons, Inc., New York, 

1957. 
Zimmermann, H. J., and S. J. Mason: "Electronic Circuit Theory," John Wiley & 

Sons, Inc., New York, 1959. 



PART 2 

TIMING 



CHAPTER 5 

SIMPLE VOLTAGE SWEEPS, LINEARITY, 
AND SYNCHRONIZATION 



The preceding chapters were concerned with the shaping of voltage 
and current signals by various configurations of linear and nonlinear 
elements, e.g., differentiation, integration, amplification, clamping, clip- 
ping, and gating. No consideration was given to the origin of the 
signals applied, but as we now begin this study, we should realize that 
many of the analytic techniques previously developed are applicable to 
our paramount problem, the generation of simple waveshapes. 

In this chapter we shall concentrate our attention oh some character- 
istics of one particular waveshape, the linear voltage sweep. Ideally, the 
output voltage increases linearly with time until it reaches a predeter- 
mined final value, instantaneously returns to zero, and immediately starts 
increasing again as the cycle repeats. This waveshape finds widespread 
application: in the horizontal deflection circuit of an oscilloscope so 
that time-varying signals may be displayed against a linear time scale; 
in radar systems, for measuring the time required for the return of the 
echo signal; in television transmitters and receivers as a means of gen- 
erating the raster; and in various control systems where it is used to time 
preprogrammed functions. 

5-1. Basic Voltage Sweep. All linear-voltage-sweep circuits, regard- 
less of individual variations, have a common basic mode of operation: 
some form of exponential charging; an automatic change of circuit which 
introduces a discharge path at the proper point of the cycle; and then 
reestablishment of the charge path. Differences in sweeps usually con- 
cern the active elements controlling the points at which discharge com- 
mences and ceases. But to approximate more closely the ideal sweep, 
we are sometimes also forced to modify the charge and discharge paths. 
Thus the basic voltage sweep might well be characterized by the circuit 
shown in Fig. 5-la, which generates the waveshape of Fig. 5-16. The 
capacitor charges through Rx toward E^. When its voltage reaches some 
preset final value E f , we throw the switch to position 2, discharging C 
through Ri. Upon decaying to its initial value E i} the switch is returned 
to position 1 and the cycle repeats. 

147 



148 



TIMING 



[Chap. 5 



Before turning to practical sweep circuits, we can reach some con- 
clusions as to their general requirements from a closer examination of 
Fig. 5-la and 6. Of prime importance is the recognition that we are 
generating, not a linear sweep, but an exponential charging curve, and 
only through restricting operation to a relatively small portion of the 
total possible curve will the capacitor voltage even approximate a linear 
rise. Thus E» must be made much higher than the desired output swing 
E r Ei to ensure reasonable linearity. Secondly, we note that the retrace 
time, a direct function of Rz, becomes insignificant only when the dis- 
charge time constant is very small. Making Ri zero obviously reduces 
the discharge time to zero, but other circuit considerations may negate 
this choice. Therefore R2 should be chosen as small as possible, con- 
sistent with any limitations imposed. 












Ebb 


^_i::_™."_ 


kE, 

>-o(t) 

Ei 


/ 


y 


v\ 


! ! ^_ 





< — 


T,— 


Xr 2 J ' 



Fig. 5-1. (a) Basic voltage sweep circuit; (b) output sweep voltage. 

Finally, we can write the equation of the capacitor voltage during 
either portion of the cycle and, by substituting the known final value E/, 
solve for the sweep duration. During the voltage rise, when n = R X C, 



Thus 



Tx = ti In 



E» - (Etb - Ede-'i'i 
En, — Ei 



(5-1) 



Em — Ef 

The capacitor discharges from Ef toward zero with a time constant 
r 2 = R2C, and consequently the sweep recovery time, the time required 
to fall to Ei, becomes 

(5-2) 



r s = r 2 lnfj 



The only remaining unknown is the mechanism of circuit switching, 
which must be evaluated in terms of the physical characteristics of the 
particular device employed. 

5-2. Gas-tube Sweep. An extremely simple sweep-voltage generator 
(Fig. 5-26) makes use of the double-valued characteristics of a cold- 
cathode gas tube to periodically discharge the sweep capacitor. As 



Sec. 5-2] 



SIMPLE SWEEPS AND SYNCHRONIZATION 



149 



long as the anode voltage remains below the ignition potential, E f of 
Fig. 5-2o, it supports only a very limited leakage current flow; for all 
practical purposes the tube is an open circuit. However, once the 
voltage rises to E f and the tube fires, the terminal voltage drops to a lower, 
almost constant value E . Both E f and E Q depend on the particular gas 
used. The tube current is now primarily limited by the external circuit 
and C begins discharging through the path established by the ionized gas. 
The discharge path will be maintained only as long as the current 
flow is sufficient to sustain ionization. As C discharges, the tube current 
drops, tracing the dashed path of Fig. 5-2a instead of retracing the orig- 
inal curve. Because we are interested in generating a repetitive wave- 
shape, we must ensure the existence of circuit conditions that will 
extinguish the tube, preferably rapidly. 



\" / 



J? 2 



— — — — — 1 o 



(a) (b) 

Fig. 5-2. (a) Gas-tube volt-ampere characteristics; (6) gas-tube sweep circuit. 

The mechanism of the gas-discharge extinction, the recombination 
of ions previously produced by the large forward-current flow, requires 
some small finite time. It is impossible to state the deionization time 
in simple terms, since it depends on the past history of the tube, i.e., on 
the peak current on ignition and on the time rate of current change 
during the capacitor discharge. This time varies from tube type to 
tube type and even changes in any given tube during its life; it usually 
ranges between 0.5 and 2 /usee. 

In order to simplify the calculations, we find it convenient to assume 
that the tube voltage remains unchanged during the whole discharge 
interyal. Additionally, a fictitious extinction current I bx might well be 
postulated as a means of representing the complete, complex extinction 
phenomena. Thus the following statement suffices: when the tube 
current drops below I bx , the tube goes out. I bx depends on the tube 
type in much the same manner as does the extinction time, with its value 
generally falling between 25 and 250 n&. 

During capacitor charging, the nonconducting tube can be omitted, 
leaving an equivalent circuit identical with that of Fig. 5-la (switch 



150 



TIMING 



[Chap. 5 



in position 1). Therefore the rise time is identically given by Eq. (5-1). 
However, evaluation of this equation is contingent on finding the still 
unknown initial value E t . Referring to the output waveshape (Fig. 5-36), 
we see that the initial value of the rise corresponds to the final value of the 
decay, i.e., to the point at which the tube extinguishes. 



■ — vw 



-*>A/\ — I 



-=-£„ ?fcc =rE, 




(a) 



Fig. 5-3. (a) Discharge circuit of gas-tube sweep; (b) gas-tube sweep waveshape. 

The circuit enters the region of operation defined by Fig. 5-3o with the 
capacitor voltage at E/ and leaves it when the output drops to E,. Imme- 
diately upon the tube's firing, C starts discharging toward the Th6venin 
equivalent voltage across its terminals. 



Et = 



Ri 



Tti -|- X12 



E + 



/L2 



Ri + R 



Eu 



(5-3) 



Extinction occurs when the tube current falls to h x , which corresponds 
to a capacitor voltage of 



e c (T 2 ) = Ei = E + I hx Rt 



(5-4) 



Since we now know the initial, final, and steady-state voltages in this 
region, the decay time becomes 



Et 



E, 



where 



Et — (E + IbxRi) 
r 2 = (fii || R*)C ^ RiC 



(5-5) 



Normally, R2 serves to limit the maximum tube current to a safe 
value and is very much smaller than Ri. Reexamination of Eq. (5-3) 
indicates that under these circumstances the Thevenin equivalent voltage 
will be only slightly larger than E„. The second term of Eq. (5-4), 
denning the initial voltage Ei, will generally be almost negligible compared 
with the first term as a consequence of the small values of both h x and Ri. 
E may vary from 10 to 50 or 100 volts, depending on the tube type, but 



Sec. 5-2] 



SIMPLE SWEEPS AND SYNCHRONIZATION 



151 



IbzRi is usually somewhere between 25 and 250 mv. The steady-state 
discharge voltage, the initial voltage, and the tube's extinction voltage 
are so close together that we might as well approximate them by the 
single value E . We conclude that the discharge curve will constitute 
virtually the complete exponential, requiring roughly four time constants 
for completion. 

T 2 ^ 4iJ 2 C (5-6) 

Occasionally, the deionization time may be longer than the time given by 
Eq. (5-6), and if it is, it will predominate. 

A glance at Fig. 5-3<x discloses that during ignition part of the tube 
current is contributed by Ebb. If this component is greater than I bt , 
the tube will never extinguish. To prevent continuous ignition, the 
inequality of Eq. (5-7) must be satisfied. 

h = RT+R* < h * (5 " 7) 

Equation (5-7) sets the lower limit of R x necessary for guaranteed repeti- 
tive sweep generation. Since Ri >5> Rz, 

Ebb — E 



Ri > 



(5-8) 



In practice, the circuit may operate properly when Ki is from one-half 

to one-quarter of the limiting value given by Eq. (5-8). This apparent 

inconsistency arises because h x is only a convenient fiction approximating 

the whole discharge phenomenon 

and a steady-state current several 

times as large may still allow 

extinction. 

Limits must also be imposed on 
Ri\ if too large, the discharge time 
becomes excessive, and if too small, 
the peak current on ignition may 
exceed the tube's rating. There- 
fore it must satisfy 



R2 > 



Ef — Eg 



(5-9) 



Fig. 5-4. Avalanche diode characteristic. 



where Ib, ml is the maximum allow- 
able tube current. 

Several semiconductor devices have two-terminal volt-ampere char- 
acteristics quite similar to that of the gas tube. Among these are the 
avalanche diode, the unijunction diode, and the p-n-p-n transistor. One 
typical characteristic is shown in Fig. 5-4, and it should be compared 



152 



TIMING 



[Chap. 5 



with Fig. 5-2a. A somewhat more detailed discussion of these devices 
will be deferred until Chap. 11, at which point some simple sweeps will 
again be treated, but from a different viewpoint from that expressed 
here. However, it is important to note that similar volt-ampere charac- 
teristics mean similar circuit behavior and therefore these semiconductor 
devices may be used to replace the gas tube in the sweep of Fig. 5-26, 
especially in low-voltage applications. 

6-3. Thyratron Sweep Circuits. The sweep of Sec. 5-2 has many 
inherent disadvantages: among them, that its sweep amplitude is small in 
comparison with the ignition potential and depends solely on the non- 
controllable firing point and sustaining voltages; that synchronization 
cannot be effected through injection of external signals; and finally that 
the relatively long recovery time precludes use for generation of short- 
duration sweeps. All but the last of these drawbacks are overcome in 
one practical sweep (Fig. 5-5) by employing a thyratron in place of the 




Fia. 5-5. Thyratron sweep circuit. 

cold-cathode gas tube. Except for the control afforded by the grid, this 
circuit performs identically with the sweeps discussed in Sees. 5-1 and 5-2, 
and consequently all equations previously derived are applicable. 

Where one curve sufficed to define completely the operation of the 
cold-cathode gas tube, two are required for the thyratron because of its 
additional control element: one expressing the grid-voltage-firing-point 
relationship (Fig. 5-6a), and the other, the plate volt-ampere character- 
istic (Fig. 5-6&). The thyratron grid is a massive structure, physically 
situated so that it almost completely shields the cathode from the 
influence of plate voltage. A negative grid voltage establishes a potential 
barrier, preventing electron travel from the cathode to the plate, cutting 
off the tube. As the plate voltage increases, its influence on the potential 
distribution in the interelectrode space between grid and cathode increases 
proportionately. Eventually the potential barrier is lowered enough to 
allow high-energy electrons into the grid-plate region, where they collide 
with gas molecules, ionizing them. This value of plate voltage is called 
the firing point of the tube. Referring to Fig. 5-6a, we see that the more 
negative the grid, the higher the plate voltage necessary for ignition. 



Sec. 5-3] 



SIMPLE SWEEPS AND SYNCHRONIZATION 



153 



Upon firing, the plate voltage drops to E , and as in the cold-cathode 
gas tube, the current increases until limited by the external circuit. 
Since the drop across the thyratron is relatively small (10 to 25 volts) 
in comparison with the firing voltage, it generates a large-amplitude 
sweep. 

Once the tube ignites, the positive ions are attracted by the most 
negative element in the tube, the grid. They form a sheath on the 
grid structure, effectively insulating it from the tube and preventing 
further control. The grid current that now flows is limited to a safe 
value by choice of the external resistor R„. Breaking the plate con- 
nection or reducing the tube current below h x extinguishes the tube, with 
the grid now regaining control. 




15<T 



(b) 
Fig. 5-6. (a) Thyratron firing and (6) thyratron plate characteristics. 

Direct algebraic solution of thyratron circuits necessitates the algebraic 
expression of the plate firing-point characteristics. The simplest 
representation is the dashed line shown in Fig. 5-6a. Its equation is 

E, = mE c + Et &£ -9E C + 5 (5-10) 

where E c < and E/ > E a . In this equation m is the slope and E 
the intercept. Substituting Eq. (5-10) into Eq. (5-1), the sweep time 
becomes 

En, — E 



Ti = ti In 



(5-11) 



Eu - (mE c + Ei) 
With a supply of 250 volts and a bias of — 10 volts, the sweep lasts for 

Ti = Tiln io^"§§ = a415Ti sec 

Under the above conditions, the total sweep amplitude is 80 volts 
(Ef — E t ) out of a maximum possible swing of 235 volts. Even over 



154 



TIMING 



[Chap. 5 



this large fraction of the charging curve, the sweep approximates a 
straight line reasonably well. 

The sweep period, a linear function of the time constant R\C, is varied 
by changing C in steps for coarse control and by changing R\ continu- 
ously as a fine adjustment. In each range the minimum resistance must 
satisfy Eq. (5-8), and therefore R\ usually consists of a fixed resistor 
of the minimum value in series with a potentiometer. 

Minor modification of the free-running sweep of Fig. 5-5 converts 
it into a single-shot generator (Fig. 5-7), which produces a single sweep 
upon each application of an input trigger. In its normal state the plate 
voltage is limited to E b by the plate catching diode D x . E c sets the firing 
voltage well above this value. The injection of a positive trigger at the 
grid momentarily lowers the ignition potential below Eb, and the tube 
immediately fires. C now discharges, the tube extinguishes, and C 




(a) "» (6) 

Fig. 5-7. Single-shot sweep generator and output waveshape. 

starts recharging, only now generating the single sweep shown. Because 
sweep starts after the discharge is complete, unless the delay introduced 
is small, a portion of the signal we wish to observe may not appear on the 
oscilloscope. 

5-4. Sweep Linearity. Since our announced objective is the gener- 
ation of a linear sweep, we should have a means of expressing the linear- 
ity, or departure from linearity, as a measure of the sweep quality. This 
entails making a comparison between an ideal and the actual sweep, 
with any divergence representing the nonlinearity. Three possible 
choices are presented in Fig. 5-8. The first is a straight line drawn tan- 
gent to the sweep at the origin (Fig. 5-8o) ; the second is a compromise, 
intersecting the sweep at the point that results in equal deviation above 
and below the line (Fig. 5-86) ; the third line connects the end points of the 
sweep (Fig. 5-8c). 

Depending on the particular application of the sweep, any one of the 
three will be eminently satisfactory as a basis for comparison. When- 
ever the sweep time duration becomes important, the sweep is usually 
synchronized by an external reference signal which constrains its end 



Sec. 5-4] 



SIMPLE SWEEPS AND SYNCHRONIZATION 



155 



points (Sec. 5-5). Under these conditions it seems reasonable to calculate 
the amplitude deviation from an ideal sweep connecting the two restricted 
ends. Referring to Fig. 5-8c, the deviation is 



«(*) = e„(0 - «,(*) 



(5-12) 



The ratio of maximum deviation to sweep amplitude, expressed as a 
percentage, is defined as the sweep nonlinearity. 



NL = ^ 100% 



(5-13) 



where E. = E t — E { . 




Fig. 6-8. Various methods of defining Bweep nonlinearity. 

By shifting the axis and writing all equations with respect to E it the 
equation of the linear sweep becomes 



e t (t) = kt 
and that of the exponential sweep 



E.. 



«.(*) = (En - E t ){X - e-"") 



(5-14) 



(5-15) 



When the limits of E, and h are substituted into Eq. (5-15), the coefficient 
can be expressed in terms of the known sweep quantities. 



Eti 



E t - 



E. 



1 _ e-tjrt 



(5-16) 



The first two terms of the power-series expansion of the exponential 
[Eq. (5-17)] represent its linear approximation, and the remaining higher- 
order terms, the deviation from linearity. Because we are interested 
in an almost linear sweep, the contribution of the higher-order terms 
must be kept to a minimum, though using only a small portion of the 
total exponential curve. Therefore t/r\ will always be much less than 



e -'/'. = i _ 

ri 



156 timing [Chap. 5 

unity and we can assume that the complete nonlinearity is due to the 
square term of Eq. (5-17). 

Substituting Eq. (5-16) into (5-15) and using the first three terms 
of the expansion in both numerator and denominator, the equation of the 
exponential sweep becomes 

(A ~ TlE ' \L _ I ^lYl 

6cW = «,(1 - h/2r0 [r» 2 \ T1 ) J 

and since ti/Vi <K 1 , 

The one missing piece of information, the time location of S m . t , will 
be found from geometric considerations. Any curve having only slight 
curvature, such as the exponential sweep of Fig. 5-8c, may be approxi- 
mated by the arc of some circle with large radius. The linear sweep 
becomes a chord of this circle, and the maximum distance from a chord 
to the arc, measured at any fixed angle, occurs at the center of the chord, 
at the point corresponding to ti/2. We evaluate Eqs. (5-14) and (5-18) 
at this time and substitute the answers into Eq. (5-12), with the result 

«--• GH£-A®']* 

^~E, (5-19) 

OTl 

Consequently, the percentage nonlinearity becomes 

NL = ~ 100 % (5-20) 

OTl 

Substitution of the exact period, expressed in terms of the circuit voltages 
[Eq. (5-1)], provides us with an alternative form for the nonlinearity, 
one which brings to the forefront the dependency on the voltage limits 
imposed. 

A third expression for the nonlinearity is possible when the sweep 
starts close to zero and constitutes only a small fraction of the total 
exponential- If E» » #, and En » E s , then by expanding Eq. (5-21) 



Sec. 5-5] simple sweeps and stncheonization 157 

into a series and taking only the first term, 

NL£^12.5%-Ji (5-22) 

Equations (5-20) to (5-22) verify our previous contention that good 
sweep linearity can be realized only by operating over a small portion 
of the total exponential. For less than 1 per cent nonlinearity, the sweep 
time must be restricted to 8 per cent of the time constant and its amplitude 
to 8 per cent of possible charging voltage. 

Since the other possibilities presented in Fig. 5-8 referred to the same 
sweep, with the same amplitude, end points, and time constant, any 
nonlinearity defined in their terms will differ from Eq. (5-20) or (5-21) only 
by a constant multiplier. 

5-5. Synchronization. The period of a sweep depends on many fac- 
tors: on the circuit time constant, on the voltage at which the capacitor 
discharge commences and at which it terminates, and finally, on the 
steady-state supply voltage. These parameters are never completely 
constant but vary slightly because of ambient-temperature change, line- 
voltage variation, aging of tubes and components, noise, etc. As a 
direct consequence of the random circuit changes, sequential sweep cycles 
will not have identical duration but will jitter about some average time. 
When this sweep is used for oscilloscope display, a corresponding jitter 
appears in the pattern seen on the screen. If used for timing, there will 
be uncertainty as to its exact time duration. However, by injecting an 
external control signal, the sweep time can be locked to some multiple 
of the synchronizing signal period, and when this input is derived from 
the display signal, the oscilloscope pattern will remain stationary. 

Synchronization of the sweep is accomplished by forcing the free- 
running sweep period h to some integral multiple of the signal period 
t„ that is, by changing h to nt,. In effecting the change of period, the 
sweep amplitude may also be affected, but as the amplitude can always 
be adjusted elsewhere in the system, this is relatively unimportant. 

The sweep period is controlled by varying either the initial voltage 
or the discharge point. For example, in the thyratron sweep of Fig. 5-5, 
the sync signal is injected at the grid, with its fluctuation about the 
quiescent grid point changing the instantaneous firing voltage. Since 
the slope of the plate firing characteristic [Eq. (5-10)] represents the 
"gain" of the gas tube, the effective variation at the plate will be m 
times as large as the applied grid signal. Figure 5-9 indicates the firing 
voltage in the presence of a sinusoidal sync signal. We observe that, 
because of the assumed linearity of the grid-plate transfer characteristics, 
the grid waveshape is simply multiplied, reflected into the plate, and 
plotted about the quiescent firing line. The sweep ends earlier because 



158 timing [Chap. 5 

it intersects the firing curve at a lower voltage value. As a consequence 
it fires at the same point of each cycle, whereas the free-running sweep 
(dashed line of Fig. 5-9) terminates randomly with respect to the applied 
sinusoidal signal. 

If we use an ideal free-running sweep instead of an actual thyratron 
sweep, the quantitative treatment below will be greatly simplified. In 
addition, the lack of identification with any particular circuit increases 



Csync 




Fig. 5-9. Applied synchronizing signal and its effect on sweep duration. 



Ey+Es 

E f 

E f -E. 


h-v-H 

*X 


•Esync 

/ if v f \ 

1 / 

1 / 






°< 


D 


1 / 
1 / 
1 / 

h 


-« 2t, H 


t 



Fig. 5-10. Ideal sweep synchronized by a triangular wave. 

the generality of our discussion. This perfectly linear sweep starts rising 
from zero at a rate of k volts/sec, ending at the firing voltage h sec later. 

The introduction of a sync signal of proper amplitude and period 
pulls the sweep into synchronization as shown in Fig. 5-10. This build-up 
may take one or several cycles, but eventually the sweep will start and 
end at the same point of the sync cycle. 

One obvious question facing us is, Why does synchronization increase 
the stability of the sweep with respect to random circuit changes? Our 
argument might begin with the observation that these changes result in 
sweep time instability, and since the effect produced is the same, any 
circuit variation may be represented as an equivalent small perturbation 



Sec. 5-5] 



SIMPLE SWEEPS AND SYNCHRONIZATION 



159 



of firing voltage. In Fig. 5-1 la, the slight increase of Ef by AEf increases 
a single free-running sweep period by At. The subsequent cycles are 
back to the normal time, but each is delayed by At sec. However, once 
the sweep is synchronized by terminating it at a point on the sync 
signal that has a slope of opposite sign to the sweep slope (in this case, on a 
point of negative slope), then the time perturbation rapidly damps out. 
Figure 5-116 shows how the sweep returns to its original firing point 
within several cycles of the original disturbance. 




ti ti+M Zti 2t!+&t 




Fig. 5-11. (a) Sweep perturbation — no sync signal; (b) sweep perturbation — sync 
signal present. 



The reason for the rapid damping can easily be seen when we expand 
two adjacent cycles of the sweep for a closer look at the end point (Fig. 
5-12). At the first firing point, the voltage jitter AEf produces an 
increase in the sweep period of At. Thus the second cycle starts At later, 
but because of the geometry at the intersection of the two straight lines, 
this cycle ends only At' later. The time variation is reduced by a factor 
S, at the instant of firing, where 



& = 



At 



(5-23) 



When the two intersecting curves have slopes of opposite sign as in 
Fig. 5-12, 8 will always be less than unity. The third sweep starts 



160 



TIMING 



[Chap. 5 



At' later, and at the instant of firing, this is also reduced by the factor 8. 
After n sweeps, the total time displacement from the normal time will be 
5" At and 

lim S" At = 

n— ♦ large 

Thus the sweep eventually returns to its original firing point with respect 
to the sync signal. We see from Fig. 5-12 that the steeper the slope 
of the synchronizing signal at the point of sweep intersection, the smaller 
the value of 8 and the fewer the number of cycles required for sweep 
recovery time. We further conclude that the optimum point to effect 
synchronization (with any input signal) is where the slope is the steepest, 





First cycle Second cycle 

Fig. 5-12. Expanded sweep firing point showing stability. 

such as the crossover point of a sine wave. If a choice presents itself, 
the optimum sync signal would be a square wave or pulse train where the 
sweep can terminate on a point of infinite slope. 

Suppose that the sweep intersects the sync signal at a point that has 
the same sign of slope; i.e.., both are positive (Fig. 5-13). Under these 
circumstances any slight time perturbation At increases at the point of 
firing by the factor 8' to At". 



8' 



= ^-'>l 
At 



(5-24) 



Instead of damping out, the change in sweep time now builds up geo- 
metrically as (8')" over n cycles. Eventually, as the sweep shifts its 
relative position, it intersects the sync voltage at a point of negative 
slope. If the sync signal has the proper time and amplitude relationship, 
with respect to the sweep, synchronization will now be effected. 



Sec. 5-6] 



SIMPLE SWEEPS AND SYNCHRONIZATION 



161 



6-6. Regions of Synchronization. Synchronization is not automat- 
ically guaranteed upon the injection of just any external signal. 
The sweep will not be pulled in unless the sync amplitude and period 
happen to fall within regions well-defined in terms of the sweep constants; 
regions, which we shall see below, are also a function of the sync signal 
waveshape. As an initial example, refer to Fig. 5-14, where a sym- 
metrical square wave of variable amplitude is used as the control signal. 





First cycle 
Fio. 5-13. Unstable sweep synchronization- 



Second cycle 
expanded firing region. 




Fio. 5-14. Square-wave synchronization. 



The first sweep fires when it terminates on the square wave at 3t„ and 
if the input remains unchanged, the sweep will continue to fire once every 
3 cycles. However, the increased square wave causes the second and 
third sweeps to end at nonintegral multiples of the sync signal period, and 
therefore they are unsynchronized. The fourth cycle is again synchro- 
nized, this time at the next smaller integral multiple of the sync period, at 
2t.. We might also note that any free-running sweep bounded by the 
dashed lines shown in the first sweep cycle, that is, t a < t\ < h, intersects 



162 



TIMING 



[Chap. 5 



the changed firing voltage along the same portion of the square wave 
and also fires at 3t,. 

Let us now assume that the control signal is a square wave having a 
constant amplitude of E, volts peak value. Then if its period falls 
anywhere within the two extremes given in Fig. 5-15a and b, the fixed 
free-running sweep will fire over the same integral number of sync 
cycles, over n cycles. At the one limit shown in Fig. 5-15a, the sweep 
period becomes foreshortened because a portion of the sync signal 
extends below the original firing line. Figure 5-15& illustrates the possible 
lengthening of the sweep as a consequence of the effective increase in the 
firing voltage. 



E,+E, 
E,h 



E/—E s 













— 




* 




— 


— 


D 


B 

— a 








i 
i 
i 

1 
i 
I 
i 
i 
i 
i 

EJC, 



h t 



(a) 




Fig. 5-15. (a) Minimum sync period producing synchronization; (b) maximum 
sync period producing synchronization. 

We shall first concern ourselves with expressing the lower usable limit 
of the sync period (Fig. 5-15a). Since the two sweep triangles OBC and 
ODE are similar, the ratios of their equivalent sides must be equal. 



OE BE 
OC BC 

Hence one limit becomes 

IT 



nt,,. 



Ej — E, 



E, 



K'-l) 



(5-25) 



The solution of Fig. 5-156, also by similar triangles, yields a second limit, 

t„ 



k,m». _ 1 ( , . E\ 

h n\ 1 ^ E,) 



(5-26) 



If the sync amplitude is now increased, the sweep will stay in syn- 
chronization until the corner of the preceding half cycle intersects the 
rising sweep, causing premature firing. This condition is indicated, for 
both cases, by the dashed lines of Fig. 5-15. Consequently, for any 



Sec. 5-6] simple sweeps and synchronization 163 

value of nt„ the following condition on the maximum allowable sync 
amplitude may be written 

■ni, - \bU _ E, - E. M 



or 



U 
E„,, 



E, 



= 1 - 



E f 

2w - 1 l„ 
2 h 



(5-27) 



Equations (5-25) to (5-27) are all straight lines in terms of the normal- 
ized coordinates t,/h and E,/E/. Each delineates one set of the boundary 



1.4 
1.3 
1.2 
l.l 
1.0 

0.9 
0.8 
«T 0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 






'« ma* 


















>i 
















































^smax 


















V * 






























's min 






















































































































n-4 














^ 



















0.1 0.2 0,3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 



I— 



Fig. 6-16. Regions of synchronization— square-wave input signal. 

values which limit possible synchronization. By plotting these equations 
with n as a parameter, we define the regions of synchronization for a 
square-wave input (Fig. 5-16). Any combination of periods and voltages 
lying within a region ensures proper synchronization over n cycles of the 
square wave. And to allow for possible fluctuation of sync and sweep 
voltage amplitude, and sweep period, the best combination of parameters 
lies at the center of any one region. Furthermore, the area of each 
region decreases extremely rapidly as n increases, with the consequence 



164 



TIMING 



[Chap. 5 



that it becomes very difficult to ensure synchronization over ratios 
greater than 4 or 5. The expected component variations would easily 
shift the operating point out of the small synchronization regions holding 
for large n. 

All sync signals that effectively increase and decrease the firing voltage 
by equal amounts, e.g., sine waves, triangles, sawtooths, etc., will have 
two limits of their synchronization regions defined by Eqs. (5-25) and 
(5-26). Equivalent limits for nonsymmetrical signals may be calculated 
quite easily from their geometry. 

For example, if the square wave used above is differentiated and 
the train of equal-amplitude positive and negative pulses is applied, 

then only the negative pulse will 
influence the timing. The sweep 
cannot be lengthened since it will 
always terminate on the negative 
pulse or along the original firing 
line, as in Fig. 5-17. In this case 
Eq. (5-26) reduces to 




t.„ 



1 



Equation (5-25) will still hold as 
a second limit. A new equation 
must be found to delineate the third 
boundary (Prob. 5-14). 

Equations (5-25) and (5-26) may 
also be applied to nonsymmetrical 

waves by properly interpreting the various voltage terms they contain. 

We rewrite these equations 



Fig. 5-17. Synchronization with pulses. 



t«,min _ I 1 ZZJ 1 — I 

h ~ n\ l Ef) 

tg.max _ f 1 _J_ 8p \ 

h -n\ y+ E,) 



(5-28) 
(5-29) 



where E,„ is the negative peak of the sync signal and E, p is the effective 
positive peak, both measured from the original firing line. Of course, as 
with the pulse, the sweep must be able to terminate at these voltage 
values. 

The difficult problem in finding the regions of synchronization is the 
evaluation of the third boundary. 



E,, m „ _ * /tA 



Sec. 5-6] 



SIMPLE SWEEPS AND SYNCHRONIZATION 



165 



For signals having simple geometric waveshapes, it is possible to find an 
explicit analytic solution, but for more complex signals we would have 
to turn to a tedious graphical construction. 

We can appreciate the increase in complexity by treating the case of 
triangular-wave synchronization (Fig. 5-10). Two limits are known 
[Eqs. (5-25) and (5-26)], and the third will be found from Fig. 5-18, which 
illustrates the conditions existing at the maximum possible value of the 
sync signal. If we assume that the sweep is still properly synchronized 
but on the verge of limiting, 



nt, 
h 



E, + KE., L 

E, 



(5-30) 



where K is a positive or negative constant relating the point of inter- 
section to the peak value of the triangular wave. The sweep starts 




Fig. 5-18. Triangular-wave synchronization — maximum value of sync signal. 

and ends when it intersects the sync wave M a sec before the triangular 
wave crosses the quiescent firing line. From the similar triangles ABC 
and DEC, 

KX. 
4 



M a 



(5-31) 



When the sync voltage becomes infinitesimally larger than the value 
assumed for Eq. (5-30), the sweep no longer clears the negative peak. 
It now fires prematurely upon intersecting the sync signal 3<„/4 — M a 
sec before the original firing point. Thus the following ratio now holds : 



nt. - (3J./4 - M a ) _Ej - E,, m 



E, 



(5-32) 



Substitution of Eq. (5-31) into (5-32) provides us with a second equation 
relating t,/h and J5, imml /JS/, one which, however, also includes the third 
variable K. The simultaneous solution of Eqs. (5-30) and (5-32), 



166 timing [Chap. 5 

eliminating K, completes the solution, with the final answer given in 
Eq. (5-33). 

ny* - y + (4n - 3)xy - 4x + 4z 2 = (5-33) 



where 



Equation (5-33), the equation of a conic section, defines the third 
boundary of the synchronization regions which are plotted in Fig. 5-19. 



1.4 
1.3 
1.2 
1.1 
1.0 

* 0.9 
0.8 

'. 0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
























































/ 






























'smax 




/ 


































u 




/ 




























































































































































71" 


■ 1 


























E 














































































































































































































(, 






















































































































S 


-^ 


*-. 
































s 


M 


l- 




































V 


M 








































V 


•^ 






































1— 






























* 










J 

















































































































































































































































































0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 



Fig. 5-19. Synchronization regions — triangular-wave input signal. 

The larger areas indicate that synchronization will be maintained in 
face of wider-latitude parameter variations than can be tolerated for 
the square wave (compare Figs. 5-16 and 5-19). However, a square 
wave or pulse is vastly preferable because of the fast recovery from sweep 
perturbations. 

The two waveshapes above were presented primarily in an effort 
to establish graphically the interlocking conditions which must be evalu- 
ated and satisfied for precise and accurate output control. We made 
no pretense of covering the multitudinous possible sync signals, such as 



Sec. 5-6] simple sweeps and synchronization 167 

pulses, sine waves, etc. When order of magnitude or approximate 
boundaries will serve our purposes, the replacement, for calculation, of 
complex waveshapes by simpler ones suffices. For example, a sinusoidal 
signal might be approximated by a truncated triangular wave, leading to 
regions slightly smaller than those of the triangle and slightly larger 
than those of the square wave. If we are interested in only one param- 
eter's variation, then evaluation from a sketch gives the allowable limits 
directly. 

Synchronization is employed in many circuits other than the thyratron 
sweep. In complex systems, often all waveform generators are controlled 
by one master, which is made extremely stable with respect to time 
changes. The process of synchronization imparts this same stability 
to the remaining circuits. Each free-running generator, oscillator, multi- 
vibrator, phantastron, etc., has its own mode of synchronization con- 
tingent on its individual control characteristics. Consequently, the 
individual regions will be well defined but will differ from one circuit to the 
other. When needed, they may be calculated by application of the meth- 
ods discussed in this chapter, modified as necessary to suit the particular 
problem. 

Example 6-1. A signal having a highly complex structure with many jagged peaks 
must be used to synchronize a sweep over a ratio of 2:1. In its original form, the 
input is not suitable for synchronization, but by amplifying and clipping, it can be 
converted into a square wave. And, if needed, we can differentiate it to obtain 
pulses or integrate it to obtain a triangular wave. The questions of interest are: 

(a) Which signal is best for synchronization? 

(6) At what point should synchronization be effected? 

Solution. For optimum operation the sweep should be synchronized so that the 
nominal operating point is in the center of the appropriate region. By this we mean 
that it should be located as far as possible from the boundaries, allowing the time and 
voltage to vary over the widest possible limits before the sweep becomes erratic. In 
general, the operating point which permits the greatest variation in the time ratio will 
not be the optimum one with respect to voltage changes. However, from Figs. 5-16 
and 5-19 we see that the time bounds are much narrower, and therefore we shall 
usually design the circuit for the best response with respect to time variations. 

1. For square-wave synchronization, the plot of Fig. 5-16 indicates that the widest 
changes in t,/t\ occur along a line through the apex of the region n = 2. The bounds 
are 

0.425 <^< 0.575 

and the nominal free-running sweep should be set at the center, i.e., at t, =» Q.5U. 
Assuming a constant sweep time, the sync signal can increase from 0.15E/ to 0.252?/ 
or can decrease to zero before the sweep becomes unstable. 

2. With the triangular-wave input, the maximum permissible change in U does not 
occur along a line through the apex. In Fig. 5-19 we see that at E./Ef = 0.3, 



0.35 <r< 0.53 



168 



TIMING 



[Chap. 5 



The free-running sweep should be adjusted to the center value h — 2.27*,. This 
establishes the following nominal limits on the sync amplitude from the operating 
point given above: 

0.12J?/ < E, < 0.5JJ, 

3. The regions which might be drawn for pulse synchronization (Prob. 5-14) would 
indicate that the optimum operating point should lie at 



t. 



0.375 



E, = 0.5jB/ 



Permissible time and voltage variations are between the limits 



0.25 < r < 0.5 



0.25 < -^ < 0.625 



To summarize, the percentage variations of free-running sweep time and sync voltage, 
as related to the nominal value of U and Ef, are: 





Percentage variation 




Square 


Triangle 


Pulse 


At 
U 

ae, 

E f 


±15 
-15, +10 


±20 
-18, +20 


±33 
-25, +12.5 



We conclude that pulses are best for synchronization purposes. Besides allowing the 
widest tolerances in time, their steep edges force recovery of any perturbation within 
one, or at most two, cycles. The triangular wave does permit wider latitude to the 
sync amplitude, but the difficulties involved in its generation as well as the longer 
recovery time would usually preclude its use; pulses or even the square wave is to 
be preferred. 



PROBLEMS 

6-1. We have at our disposal a cold-cathode gas tube with the following character- 
istics. The tube fires at 90 volts and maintains a constant drop of 50 volts while 
ionized. It requires at least 50 ixa to sustain conduction and would be damaged by a 
current in excess of 200 ma. Using a power supply of 200 volts, compute the mini- 
mum values of Ri and R t that may be employed. Adding a safety factor of 20 per 
cent to the resistor values found, compute the value of C necessary to give a 100-cps 
sweep. 

6-2. The circuit of Fig. 5-20 is a schematic representation of a basic current sweep. 
The switch stays in position 1, while the coil current rises from its initial value toward 
I„. At the firing-current value I/, the switch is thrown to position 2, discharging the 
coil. Once the current falls to its initial value, the switch is returned to position 1 
and the cycle repeats. 

(a) Draw the current waveshape il, indicating all time constants. 

(6) If /„ = 200 ma, /< = 80 ma, and Gi = 0.001 mho, what values must be chosen 
for // and Ri if we wish to produce a triangular waveshape? 



SIMPLE SWEEPS AND SYNCHRONIZATION 169 

(c) When the inductance is 1 henry, to what must Gi and Ri be changed if the tri- 
angular sweep period is to be 2 msec? 




Fig. 5-20 

5-3. We have a thyratron with the following characteristics: /max - 1 amp, lbx = 
100 )>&, Ef = 10 — 5e e , and E a = 10 volts. This tube is used in a circuit similar to 
Fig. 5-5, where E a = 300 volts, Ri = 3 megohms, E e = -20 volts, C = 0.001 /if. 

(a) What is the sweep frequency of this circuit? 

(6) For the optimum sweep response, what value should be assigned to i2 s ? 

(c) With R 2 as given in part 6, what is the sweep recovery time? 

5-4. A 884 thyratron has a firing characteristic which may be approximated by 
Ef = 16 — 10e e . Its maintaining voltage E a is 16 volts, and lbx is 200 pa,. Design a 
free-running sweep of 100 pseo duration that will be not more than 5 per cent non- 
linear. Use a power supply of 250 volts and the minimum value of sweep resistance. 
In the interests of simplicity, assume that the discharge time is insignificant. Specify 
all circuit values and all sweep voltages. 

5-5. We wish to use the thyratron of Prob. 5-4 in a circuit that will generate an 
approximately triangular waveshape. (The rise and fall times are identical.) The 
free-running peak-to-peak output amplitude should be 50 volts. 

(a) If the available power supply is 300 volts, and if J2i -= R t , calculate the initial 
and the firing voltage for this sweep. 

(6) Express the period as a function of the time constants and voltage values. 

6-6. Design a driven sweep which makes use of the thyratron of Prob. 5-3. The 
sweep is to have a duration of 300 /isec and an amplitude of 50 volts and is to be trig- 
gered by a positive pulse 1 volt greater than the minimum amplitude. Use a power 
supply of 300 volts and a capacitor of 10 - ' farad in your sweep. 

(o) Find suitable values for E c , R u R t , R 3 , and R t in Fig. 5-7. Express Ri and Rt 
as a ratio rather than as actual resistance values. 

(6) Calculate the minimum pulse amplitude for triggering. 

(c) How long after the pulse is applied will the sweep start? 

5-7. The circuit shown in Fig. 5-21 is a triggered sweep operated by the 20-volt 
positive input triggers. It uses an 884 thyratron whose characteristics are given in 
Prob. 5-4. 

(o) What is the minimum spacing of the trigger pulses U so that each trigger will 
initiate one complete sweep? 

(6) How long after the trigger is applied will the sweep start? 

(c) Sketch the output waveshape, giving all values and times. 

6-8. Assume that the input impedance appearing from grid to cathode of the sweep 
circuit shown in Fig. 5-21 is 50 n/t! in parallel with 1 megohm before firing. What is 
the narrowest pulse that would institute a sweep? Sketch the voltage appearing 
from grid to cathode if, after firing, the grid-to-cathode resistance falls to 2,000 ohms. 

6-9. (a) Calculate the output sweep period and amount of nonlinearity for the 
circuit of Fig. 5-22. The initial value of the sweep is 20 volts, and the thyratron fires 
when the capacitor voltage reaches 100 volts, 



170 



TIMING 



[Chap. 5 



|20v 



» 


to 


' * 


At 


* 






-*!— 




t 




5M< 



^VW-*-AAAr- 
,,20K 40K i+250v 



1° 
1 



(6) Replace the triode by a resistor that will give the same sweep time and ampli- 
tude. Recalculate the linearity and compare with the results obtained in part a. 




Fig. 5-22 

5-10. A simple thyratron sweep starts charging from E { = 20 volts toward 250 
volts, and the tube fires at E, = 100 volts. 
(a) Calculate the sweep NL. 

(6) Considering the following parameters, one at a time, find the percentage change 
in m, E lt Ef, E c , and £» which will result in a 1 per cent increase in sweep nonlinearity . 
(Hint: In each case expand the natural logarithm into a series.) 

(c) List these parameters in the order that they affect the sweep quality. Assign 
the value of 1 to the term having the most pronounced effect and give proportional 
values to each of the other terms. 

6-11. (a) Repeat Prob. 5-10 with respect to the sweep duration; i.e., calculate the 
permissible changes in the various parameters, taken one at a time, which will cause 
a 1 per cent change in the sweep duration. 

(6) By reference to other texts on gas tubes, discuss the effect of temperature 
variation on the sweep duration. Is there any way to minimize this problem within 
the expected range of 30°C change in ambient temperature? 

5-12. Derive an expression for the percentage nonlinearity of the sweep, starting 
from the straight-line representation of Fig. 5-8a. Express your answer in terms of 
the sweep period and the time constant and also in terms of the critical voltages. 

5-13. Verify Eq. (5-22). 

5-14- Plot the regions of synchronization for a train of equally spaced positive and 
negative pulses. Assume that the pulses are extremely narrow. Explain your 
answer. 

5-16. A 3-volt peak-to-peak square wave is injected at the grid of a thyratron sweep. 
The tube is biased to fire at a nominal value of 100 volts, and the drop across the con- 
ducting tube is constant at 25 volts. We can express the firing characteristic as 
E f - 20 - 10e„. 

(a) If the free-running sweep period is 500 /jsec, what range of sync frequencies will 
cause the sweep to lock in over a ratio of 2 : 1 ? How does the amplitude of the sweep 
extremes compare with the nominal free-running values? 



SIMPLE SWEEPS AND SYNCHRONIZATION 



171 



(b) When the square-wave period is 400 jisec, what range of the syno voltage will 
effect synchronization? Calculate the bounds by means of a graphical construction 
and then check the results against Fig. 5-16. 

5-16. A perfectly linear sweep has a normal period of 1.0 msec. This can be 
expected to vary by as much as ± 10 per cent because of the variations in the firing 
voltage. In the absence of synchronization, the sweep starts at zero volts and termi- 
nates at the nominal value of Ef — 100 volts. The retrace time is zero. It is 
required that the sweep be synchronized so that its period is exactly 1.0 msec despite 
the instability in amplitude. 

Three synchronization waveshapes are shown in Fig. 5-23 plotted at the firing line 
of sweep. In each, (, is 1.0 msec or some integral submultiple. 

(a) Which among these will provide the desired synchronization? 

(6) When the sweep is synchronized under the conditions specified, what is the 
possible number of cycles of the synchronizing wave over which synchronization can 
take place? 



e.i 



««2 



e,3 



-«. H i 



Ti 



T 



h>~* *!*■"* \ 15v 



15 v 



-W 



1 i 



15 v 



45 v 



Fia. 5-23 

6-17. (a) Verify Eq. (5-33) by direct solution from the conditions of triangular- 
wave synchronization. 

(6) Calculate the slope of Eq. (5-33) at x = 1 as a function of n and compare it 
with the slope of the t,, m in/ti line of the n + 1 region. 

6-18. Consider that the signal available to synchronize a sweep at 10 kc must be 
derived from a 50-kc sinusoidal oscillator. It will be clipped to a symmetrical square 
wave and may also be differentiated. Calculate the optimum free-running sweep 
period and the optimum ratio of sweep to sync signal. Justify your answer. What is 
the maximum number of cycles over which the sweep can be synchronized if the free- 
running sweep has ±3 per cent jitter? 



BIBLIOGRAPHY 

MacLean, W. R.: The Synchronization of Oscilloscope Sweep Circuits, Communicor- 
tions, March, 1943. 

Millman, J., and H. Taub: "Pulse and Digital Circuits," McGraw-Hill Book Com- 
pany, Inc., New York, 1956. 

Puckle, O. S.: "Time Bases," 2d ed., John Wiley & Sons, Inc., New York, 1951. 



CHAPTER 6 

VACUUM-TUBE VOLTAGE SWEEPS 



6-1. Introduction. Even a perfunctory reexamination of the simple 
gas-tube sweeps of Chap. 5 testifies to their inherent unsuitability for 
other than the most nonoritical applications. Generation of very-short- 
duration sawtooths, smaller than about 20 usee, was precluded by the 
circuit's excessively long recovery time. Instability, introduced by the 
dependence of the ionization potential on external factors, temperature, 
noise, etc., limited the longest sweep duration to approximately 0.1 sec. 
And only through severely restricting the amplitude could sweep linearity 
be maintained within acceptable tolerances. Consequently, when strin- 
gent specifications must be met, we are forced to look for other, better 
techniques of sweep generation. 

In this chapter each of the three sweep essentials, rise, recovery, and 
switching, will be separately treated with a view toward optimizing the 
over-all performance. The first sections are concerned with the establish- 
ment of an almost ideal, linear charging path. For this purpose, it is 
necessary to employ active circuit elements, which we shall, in this 
chapter, limit to vacuum tubes. Three methods of linearization, all 
involving feedback, are in common use: current feedback for constant- 
current charging of the capacitor, positive-voltage feedback wherein the 
needed correction term is first developed and then applied, and finally 
the effective multiplication of the time constant and charging voltage, 
through the use of negative-voltage feedback. After examining each one 
of these methods of linearization, the improvement of the sweep recovery 
time and the nature of the switching process will be considered. Later 
sections will combine the individual circuits into two practical voltage 
sweeps, the Miller integrator and the phantastron. 

6-2. Linearity Improvement through Current Feedback. If we were 
to examine the differential equation defining capacitor charging, we 
would conclude that its terminal voltage rises linearly with time only 
while being charged from a constant-current source. One method of 
approaching this ideal situation is through the application of current 
feedback, as illustrated in the sweep circuit of Fig. 6-la. Here the 
complete capacitor current flows in R K , producing a proportional volt- 

172 



Sec. 6-2] vacuum-tube voltage sweeps 173 

age drop, which, together with E, determines the grid-to-cathode voltage 
(e„t = E — i b RK)- Any change in current automatically establishes the 
conditions which act to return the current flow to its original value. 
Quite possibly some factor may cause a momentary decrease in i b , and the 
reduced current will, of course, reduce the voltage drop across Rk- But 
this increases e„*, which, by inducing a resultant increase in tube current, 
effectively opposes the original change. An increase in load current 
produces just the opposite effect, again with the tube aiding in the effort 
to maintain the charging current constant. 

This circuit, from grid to cathode, is simply a cathode follower. There- 
fore, provided that the plate voltage remains sufficiently high (maximum 
capacitor voltage limited), the grid will not draw current and we can 
assume unity gain as a reasonable first approximation. The drop across 





/<*«* 



(a) (b) 

Fig. 6-1. (o) Sweep using current feedback for linearization; (6) equivalent circuit. 

R K is identically the grid-to-ground voltage E and it logically follows 
that the charging current must be 



*& = 



Rk 



50 
70 K 



= 714 n& 



(6-1) 



In order to verify this rough approximation, we might calculate the 
TheVenin equivalent circuit across C. Referring to Fig. 6-16, 

R™ = r p + (m + 1)Rk = 70 K + 7,070 K (6-2a) 

Etk = En + nE = 300 + 5,000 volts (6-26) 

With a high-ju tube, nE » #», (m + 1)R K » r p , and Eq. (6-1) gives a 
reasonably correct answer. The actual short-circuit current Eth/Rtk 
is only 4 per cent larger. Constraints, which are considered below, 
limit the maximum output sweep voltage to a very small percentage 
of the total Thevenin equivalent value given in Eq. (6-2). In spite 
of the actual exponential charging toward E T h with the long time con- 



174 timing [Chap. 6 

stant (n + 1)RkC, the current varies but slightly over the restricted time 
range of interest. 

The plate voltage falls from En, at a rate determined by the capacitor 
charging. Substitution of the approximately constant-current equation 
(6-1) into the charge equation yields 

1 /" F 

e 2 = En - ^ / % dt = E» - R —^ t (6-3) 

Equation (6-3) remains valid until the grid begins conducting. Its load- 
ing of the constant-voltage source will change the charging current and 
adversely affect the sweep linearity. The bottoming value of plate 
voltage is found from the circuit of Fig. 6-16 by setting e gk = 0. At this 
point, the actual voltage drop across Rk is E, and since the current that 
flows through Rk also flows through r P , the lower bounding value of the 
Linear sweep output becomes 



( I + fc) 



Eu = V 1 + R J ® = 10 ° volts (6 " 4) 

The capacitor discharge mechanism is usually triggered when the out- 
put falls to some preset voltage. In a fixed duration sweep, this might 
be the bottoming value found in Eq. (6-4). If, however, the charging 
current is adjusted by varying R K , then the saturation voltage will also 
change. Decreasing R K increases the current, thus increasing the charg- 
ing rate and shortening the sweep. But any reduction in R K also 
affects the possible sweep amplitude by raising the plate bottoming 
voltage. The discharge point must correspond to the worst condition, 
i.e., the solution of Eq. (6-4) for the smallest R K . Hence, from Eq. (6-3), 
the sweep time becomes 

it- RkC E »~ E ' u (6-5) 

where, for example, when Rk.™* = 35 K, E' u = 150 volts. 

As the plate falls from 300 to its bottoming value of 150 volts, the 
capacitor charges by only 150 out of its equivalent steady-state value of 
5,300 volts. Hence the sweep nonlinearity may be found from Eq. 
(5-22). 

1 "SO 
NL - POT) 125 % - °- 36 % 

In an equivalent RC sweep, where C charges by the same amount from 
a constant 300-volt source, the nonlinearity [Eq. (5-21)] is 

NL=125%ln 300^150 = 8 - 7 % 
The improvement is quite impressive. 



Sec. 6-3] 



VACUUM-TUBE VOLTAGE SWEEPS 



175 



6-3. Bootstrap Sweep. Bootstrapping, a form of positive feedback, 
linearizes the sweep by first determining what voltage correction term 
is needed and then deriving it through acting on an already existing 
voltage. The necessary circuit modification usually follows directly 
from the analysis of the original problem posed. For example, the non- 
linear charging in the RC circuit of p 
Fig. 6-2 is a consequence of the 
dependency of the loop current upon 
the capacitor voltage. 



E' - e e 
R 



(6-6) 






Fig. 6-2. Simple RC sweep. 



If, somehow, we could cancel the 
effects of e c , through the adjustment 
or replacement of one of the circuit components under our control, then 
the desired linear charging would be assured. One obvious course is to 
replace E' by E + e c . This required drive can be developed quite 
easily by amplifying the capacitor voltage and connecting the charging 
resistor to the amplifier output instead of to the battery. Because of the 
special charging requirements, the output must have a d-c level of E. 
Thus it follows that Fig. 6-3 represents the general form of the bootstrap 
sweep. In this case positive-voltage feedback is employed to increase 
the effective charging source voltage at the same rate as the increase 
in the sweep voltage e c . 



ft 



l_ 



C± + * 



R 
-VW- 



>E 



E+Ae,, 



A<1 




"0 «o *1 «a' * 

Fio. 6-3. General bootstrap sweep and output waveshapes. 



Constant-current charging of C is contingent on the amplifier gain A 
remaining unity. Any variation is immediately reflected as a change in 
the output waveshape and sweep period (Fig. 6-3). This dependency on 
A may, if serious, prevent the use of the bootstrap circuit for all but non- 
critical applications. 

The sensitivity of sweep duration with respect to the voltage ampli- 
fication may be interpreted quantitatively by starting from the equivalent 
circuit controlling the charging current flow. Referring to Fig. 6-3 and 



176 TIMING 

writing the input node equation yields 

E + Ae c — e c 



R 



E 
R 



(1 - A)e c 
R 



[Chap. 6 



(6-7) 



The right-hand side of Eq. (6-7) represents the circuit as seen by the 
capacitor: it consists of a constant-current generator in parallel with its 
own internal admittance (Fig. 6-4). 

We might observe, from Fig. 6-4, that when A < 1, the charging of C 
is exponential rather than linear. The final steady-state voltage, deter- 
mined by setting i c = 0, is E/(l — A), and by inspection of the circuit, 
we see that the charging time constant is CR/(l — A). Therefore the 
sweep voltage may be expressed as 



«.(0 = 



E 



1 - A 



(1 



-RC 1 ) 



(6-8) 



However, when A = 1, G Q becomes zero and C is charged directly from 
the ideal constant-current source. 



A = 1 



_E_. 
e °~ RC 1 



(6-9) 



Suppose that a change in either tube or circuit parameters makes A 
slightly greater than unity. Under these circumstances G„ becomes 

negative and the terminal voltage 
increases exponentially toward in- 
finity. The equation defining this 
region of operation is the same as 
Eq. (6-8). But A > 1 means that 
the time constant is now negative 
and that the system's single pole 
will move along the real axis into 
the right half plane. Since a physi- 
cal device cannot handle infinite-amplitude signals, any slight disturbance 
immediately drives the amplifier toward saturation. 

In order to see how the sweep time depends on A, we first calculate 
the sweep duration from Eq. (6-8). 




Fig. 6-4. Bootstrap equivalent circuit. 



t' = 



RC 



1 



In 



E 



E - (1 - A)E, 



No generality is lost by choosing a convenient termination point, one 
which will keep A in the forefront and at the same time eliminate super- 
fluous voltage terms. The greatest simplification occurs by setting 



Sec. 6-3] 
E, = E: 



VACUUM-TUBE VOLTAGE SWEEPS 



ta 



RC . 1 

r=-i ln A 



177 



(6-10) 



As a basis for comparison, the duration of the linear sweep of the same 
amplitude is evaluated from Eq. (6-9). 



h = RC 

And from Eqs. (6-10) and (6-11), 

ta = 1 

h 1 - 



A A 



(6-11) 



(6-12) 



For A = 0.95, 5 per cent below the optimum value of unity, 

j- = 1.082 

or the sweep time is now 8 per cent longer than the perfectly linear sweep. 
When A — 1.05, the sweep duration is reduced to 0.976<i. The separa- 
tion from the linear sweep with respect to changes in A becomes even 




Fig. 6-5. Practical bootstrap amplifier. 

more pronounced as the sweep amplitude increases. From the above 
discussion, the only possible conclusion drawn is that the gain of the boot- 
strap amplifier must be highly stabilized if we are to avoid sweep-duration 
instability. But even with the high degree of sensitivity of the sweep 
period to changes in A, the sweep linearity and maximum sweep ampli- 
tude are still superior to what it is possible to achieve with a simple RC 
network. 

A practical, direct-coupled amplifier for bootstrapping applications 
appears in Fig. 6-5. Negative-voltage feedback, applied through the 
resistor network R a and Rf, both sets and stabilizes the over-all positive 



178 



TIMING 



[Chap. 6 



gain. This amplifier's gain varies by less than 0.01 per cent, with regard 
to expected tube and parameter chaiiges from its nominal unity gain 
value. Bz controls the output voltage level, and thus the charging 
current: it might also serve as the fine sweep control. An obvious 
circuit disadvantage is the need for two power supplies which, for optimum 
gain stability, must be highly regulated. 

6-4. Miller Sweep. The Miller sweep, which utilizes negative-voltage 
feedback for sweep linearization, poses no problems as to poles in the 
right half plane. And, in addition, it generates a sweep virtually inde- 
pendent of the amplifier gain. Consider the basic circuit of Fig. 6-6a, 




(a) 




Fig. 6-6. (a) Basic Miller sweep; (fc) Miller sweep — equivalent-circuit representation. 

where the amplifier has a forward voltage gain A and an output imped- 
ance R,. If, as was done for the bootstrap, we write the current equation 
at the input node, 

. _ e,- — Ae,j 
%i ~ R, + 1/pC 

We now, by solving this equation, find the input impedance, 



„ e ( _ R. 



+ 



i 



pC(l - A) 



(6-13) 



Equation (6-13), the Miller effect, expresses the effective multiplication 
of the feedback capacity and reduction of resistance due to the negative- 
voltage feedback. With current feedback (the circuit of Fig. 6-1), the 
opposite effect appeared. It follows, from Eq. (6-13), that the input 
circuit may be represented as shown in Fig. 6-65. If the current flow 
from the output back to the input is kept small, then the drop across R, 



Sec. 6-4] vacuum-tube voltage sweeps 179 

due to the feedback component may be neglected. R controls the charg- 
ing rate as well as limiting the feedback-path current flow; it is usually 
very much larger than R,. We have thus effectively isolated the input 
and output circuits, allowing an independent solution for the input 
voltage. It may then be reflected into the output by multiplying by 
the circuit gain e = Ae t . 

Since we are interested in generating a linear sweep, charging must 
be limited to a small portion of the total exponential. The input circuit 
will be treated as an integrator with the reflected resistance R,/(l — A) 
contributing an additional small constant term. 

R, 1 /"' 

e% = -J5 — : — ttA j-t E H — 7 r — / E dt 

R - + m - A) (« + r #i)«'-*>i 

And the amplified input voltage appears at the output: 

. AR.E AE 

6 ° = Aei = B. + g(l - A) + -' — ' (6 ' 14) 



(« + A) 



C(l - A) 



If the amplifier gain is a very large negative number, then (1 — A) ^ | A\ 
and A cancels out in the sweep term of Eq. (6-14). This necessary 
condition must be satisfied in order to make the sweep relatively inde- 
pendent of the amplifier parameters. As a further result of the large 
gain, R » R,/(l - A) and 

_ R, „ E 
e '--R E -RC t 

If the much more stringent requirement that R ^> R, is also satisfied, 
then this equation may be reduced even further: 

eo^-^t (6-15) 

Equation (6-15) has the same form as the charging equation derived 
for the bootstrap [Eq. (6-9)]. But while the linear charging in that 
circuit was critically dependent on the value of the amplifier gain, this 
is not true in the Miller sweep, where the only requirement is a large 
negative gain. 

An amplifier with a gain of —100 and source voltage of 300 volts 
generates a sweep of 100 volts, while the input only changes by 1/A times 
as much, or by 1 volt. Since this is a very small portion of the possible 
300-volt charging curve, the sweep will be extremely linear; nonlinearity 



180 timing [Chap. 6 

of better than 0.1 per cent is easily achieved, and with a very-high-gain 
amplifier even 0.01 per cent nonlinearity becomes possible. 

Triode Miller Sweep. The procedure used in calculating the circuit 
waveshapes and in subsequent verification of sweep linearity is outlined 
below for the triode Miller sweep of Fig. 6-7o. The solution follows 
in time sequence the behavior of the circuit. In each region we draw the 
model holding and use it to evaluate time constants and boundary 
conditions. 

Along with the general solution outlined below, we shall consider 
the specific components of Fig. 6-7 to illustrate the order of magnitudes 
involved. 

g P, p R '- ,,r - 



± 





rntrr 



T 




(a) (b) 

Fig. 6-7. (a) Triode Miller sweep; (6) model holding in the active region. For the 
purposes of illustration, the following typical values are assigned to this circuit : in = 60, 
r„ = 10 K, r c = 400 ohms, Rl = 50 K, R = 1 megohm, C = 0.001 M f, and En = 250 
volts. 

Before closing the switch at t = 0, the grid, returned through R to En,, 
is conducting. Since the 400-ohm grid resistance is much less than R, 
the capacitor charges to Eu, with the polarity indicated. The grid is not 
actually at zero but at some small positive value, usually a fraction of a 
volt, as given by Eq. (6-16). 

400. 



E c (0 ) = -5 E» — 



10 6 



250 = 0.1 volt 



(6-16) 



Once the switch closes, the first question of interest is, What is the 
new value of grid and plate (point b) voltage? Our qualitative argument 
as to the circuit behavior follows. The voltage across the capacitor 
cannot change instantaneously; therefore any change at the plate must 
also appear as an equal change in grid voltage. When the switch closes, 
the grid either rises, falls, or remains constant at its slightly positive 
value. We shall first assume that it remains unchanged. The plate 
current corresponding to the positive grid voltage is very large, 
causing a large voltage drop at point b which, coupled by the capacitor, 



Sec. 6-4] 



VACTJTJM-TUBE VOLTAGE SWEEPS 



181 



appears at the grid and cuts the tube off. Hence a contradiction arises, 
and the original assumption must have been incorrect. 

This contradiction may be resolved by observing that only a small 
change of plate voltage may be tolerated without cutting off the tube. 
The plate current flow, after switching, must be small, and we conclude 
that the tube is driven almost to, but not quite into, cutoff. 

The operation in the active region, including the initial drop at both 
the plate and grid, may be calculated from the model which holds 
immediately after switching. In Fig. 6-76, the complete circuit, seen 
looking into the plate of the tube, was replaced by its Thevenin equiv- 
alent. E h , is the saturation value of the plate voltage (e„* = 0), and A 



-=-£n —I— 



=T=C(1-A) 



1 ! 



I- A 




(a) 



(b) 



Fia. 6-8. Triode Miller-sweep models, (a) Grid circuit; (b) plate circuit. 

is the voltage gain in the active region. From the unloaded plate-circuit 
model, 

-mKl „ _ r p w (617) 



A = 



r P + R L 



Eh, 



r p + Rr, 



Eh 



With the parameters given, the equivalent plate-circuit resistance 
(R, = r p || R L ) is only 8.3 K, the gain is —50, and E b , = 41.7 volts. 

Following the argument presented with respect to the general Miller 
sweep, the complete timing can be reflected into the grid circuit (Fig. 
6-8o). The only question remaining unanswered is, What is the initial 
voltage across the reflected capacity C(l — A)? Just after switching, 
the loop voltage from grid to ground can be found from Fig. 6-76. 

#0(0+) = -#„, + I c (0+)R, + AE C {Q+) + E„, 

Solving for the grid voltage yields 



E c (0+) 



— Ew, -\- Eb, 
1 - A 



+ 



R. 



1 



/c(0+) 



(6-18) 



The first term in Eq. (6-18) is the voltage across C(l - A) immediately 
after the switch is closed. 

Since the current through R remains invariant across the change in 
models from Fig. 6-76 to Fig. 6-8a, the second term of Eq. (6-18) can be 



182 timing [Chap. 6 

identified as the drop across the reflected source resistance. The current 
which charges C(l — .A) is 

E» - B c (0+) 



/c(0+) 



R 



but E c (0 + ) must he within the small grid base during the entire sweep. 
Hence En, ;» E c and 

Ic £* ^ (6-19) 

The almost constant charging current indicated by Eq. (6-19) proves 
that the sweep produced will be highly linear. 

By substituting the terms from Eqs. (6-17) and (6-19) into Eq. (6-18), 
we can also express the initial grid voltage as 

^ 0+ > = - r, + Of; 1)B L *» + 1^2 f <**» 

In this example, the reflected source resistance is only 160 ohms, 
which is very much less than the 1-megohm timing resistance. The 
second term of Eq. (6-20) is almost completely negligible; it is only 
0.04 volt. With a high-/* tube the significant first term of Eq. (6-20) 
may be approximated as 

E c (0+) = — J?£ = -4.1 volts (6-21) 

M + 1 

We might note that this is only slightly above the grid cutoff voltage of 
— Etb/n ( — 4.2 volts). This verifies our previous contention that the 
tube is driven to the verge of cutoff. 

After the initial drop, the grid starts charging from E c (0 + ) toward 
Em, with the long time constant 

n = C(l - A)(r + j^-t) = RC(l - A) (6-22) 



With the constant charging current assumed in Eq. (6-19), the small 
portion of the exponential rise used may be approximated by 

«.(«) = E c (0+) + RC{ f *_ A) t (6-23) 

The grid rises at a rate of Ej»/i?C(l — A) volts/sec from its initial value. 
The abrupt change in voltage at the grid, as the tube turns on, is 

A# cl = E c (0+) - E c (0~) 

Since the plate and grid are coupled by C, 

E t (0+) = Ea, - |A£ c i| = 246 volts 



Sec. 6-4] vacuum-tube voltage sweeps 183 

The plate waveshape will simply be an amplified version of the signal 
appearing at the grid. It follows that the plate will fall A times as fast 
as the grid rises. 

and since A is a large negative number, the plate rundown is independent 
of the gain. 

*(*) =-&(()+) -|g* (6-24) 

The next change in the circuit state occurs when the grid reaches 
zero and starts conducting. From Eq. (6-23), the linear grid rise of 
E e (0+) volts takes 

h= ~ E ° { ° +) ti sec (6-25) 

EtVb 

With the values given in Fig. 6-7, the sweep lasts for approximately 
800 jxsec. Only a small fraction of the 50-msec grid time constant is 
used for the linear plate rundown, and over this interval the sweep 
nonlinearity is 0.16 per cent. 

The plate voltage corresponding to zero grid voltage Eta may be 
found in several ways. For example, the sweep time found from Eq. 
(6-25) can be substituted into the plate-voltage equation. A somewhat 
more accurate method is to solve the plate-circuit model of Fig. 6-86 with 
e c = 0. 

Ebo = IcRi + Et, = -p- Ebb + Eb, (6-26) 

In the case where R, <3C R, the bottoming voltage is almost exactly E it . 
In the example cited, Ebo = 43.7 volts, only 2 volts above Eb,. 

The several answers for Ebo will not be in exact agreement because 
of the approximations made in the course of the analysis. The value 
found from the model is more nearly correct since it comes from^the basic 
circuit instead of being the product of many intervening steps. 

After the end of the linear rundown, the grid's conduction does not 
remove the tube from its active region; it only reintroduces r c from grid 
to ground in Figs. 6-76 and 6-8a, changing the time constant and the 
steady-state voltage. From the Thevenin equivalent of the grid circuit, 
the new values are 

E„2 = •=; Ebb = E C (Q~) 

K (6-27) 



T2 - C(l - A) 



(-+'fm 



184 



TIMING 



[Chap. 6 




In this region the 160-ohm contribution due to the reflected source 
impedance becomes significant compared with the 400-ohm grid resist- 
ance. The new time constant of 28 /*sec is obviously much smaller than 
n, and the rise rate is much faster. The positive grid resistance is 
introduced into the circuit at zero volts at a point where it cannot 
abruptly change the voltage anywhere within the timing capacitor loop. 

Consequently the sweep will enter 
the new region smoothly, without 
any jumps appearing in either the 
grid or plate waveshapes. 

As the grid continues rising, the 
plate falls proportionately, finally 
bottoming at Et, mill . We evaluate 
this voltage from the plate-circuit 
model (Fig. 6-86) by setting e c = 0.1 
volt. Even in the positive grid 
region, the ratio of the change in 
plate voltage to the change in grid voltage is the circuit's amplification. 
Thus E b , min might also be found by writing 

E b , min = E b0 + AE e (0~) = 38.7 volts 

Once the tube bottoms, the sweep is absolutely stable and remains 
in this state until external conditions force a change. When we open the 
switch, we remove the active element and must again draw a new model, 
this time to define the behavior during recovery. Figure 6-9 illustrates 
the conditions prevailing. 

Just before, and therefore just after, switching, the voltage across C 
is .Et,„i». The net change in the loop voltage will appear across the total 
series resistance. It will divide proportionately across Rl and r c , but 
remember that the jump at both the grid and plate must be equal. The 
grid voltage jumps by &iE c , from E C (Q-) to a new value, £„,„«. 



Fig. 6-9. Circuit model holding during 
recovery. 



E c . 



= -Ec(O-) + 



r c + Rl 



[E» 



E, 



b.min 



E,(0-)] 



Rl 



(Ett - £».„,„) = 1.69 volts 



(6-28) 



Point b rises by the same amount. Recovery now proceeds with the time 
constant 

t 8 = (Rl + r c )C S RlC (6-29) 

toward the original steady-state voltages, E» and E c (0~). 

The complete waveshapes, with all voltages, times, and time constants 
indicated, are presented in Fig. 6-10. 



Sec. 6-4] 



VACUUM-TUBE VOLTAGE SWEEPS 



185 



Complete recovery requires 4r 3 sec, or 200 usee, which is on the same 
order of magnitude as the plate's linear rundown. Unless we do some- 
thing to shorten this time (Sec. 6-5), an unreasonably long interval must 
elapse between sweeps. 




Fig. 6-10. Triode Miller-sweep plate and grid waveshapes (not to scale). 

Voltage Control over Sweep Time. External voltage control of the sweep 
duration may be incorporated by coupling point 6 (the plate) of the 
Miller sweep through a plate catching diode to the control voltage 
E (Fig. 6-11). With the circuit in the normal, unexcited state, the 
switch is open and the diode con- 
ducts, maintaining point b at E volts. 

The grid voltage just after switch- 
ing can be found by following the 
procedure previously discussed, with, 
however, an initial voltage of E 
across C. If R » R L \\ r P and the 
contribution due the reflected resist- 
ance can be ignored, we obtain 



E c (0+) 



TpE,,,, - (r p + R L )E 



(6-30) 




Fig. 6-11. Circuit for voltage control of 
sweep period — Miller sweep. 



r p + ( M + 1)R L 

Equation (6-30) expresses the linear 
relationship existing between E c (0+) and E. Since the sweep time is 
determined by the amount of the initial grid drop, it follows that the period 
is a linear function of E. However, E must be restricted to the range 
from Evb to Eh.. When E reaches its maximum value Em, Eq. (6-30) 



186 



TIMING 



(Chap. 6 



reduces to Eq. (6-21), and when E is set equal to E b „ E c (0 + ) becomes, as 
expected, zero and the sweep time is also zero. 

The initial drop in plate voltage, which brings it below E, back-biases 
the diode, effectively removing E from any further circuit control. 
Rundown and bottoming proceed as before, to the same final values, 
except that the starting point is from a lower initial voltage and the sweep 
ends sooner. 

During the recovery of the plate toward Ebb, the diode conducts 
when E b reaches E, stopping the recharging of C; the recovery time is 

reduced to some percentage of its pre- 
vious value. And often, for just this 
important reason, a diode is connected 
10 K between the plate and a voltage divider 
from Ebb to ground. 



+300 



500 K 




20 K 



Fig. 6-12. Miller-sweep circuit for 
Example 6-1. 



Example 6-1. Figure 6-12 shows a Miller 
sweep circuit which incorporates a plate 
bottoming diode, biased to conduct when 
the plate falls to 200 volts. Thus the 
duration of the linear sweep depends on the 
plate rundown rather than on the grid 
voltage rise. Furthermore, in this circuit the 
source impedance is relatively large and 
cannot be neglected. 



The initial value of the grid voltage is 



£c(0-) - 



1 K 

500 K 



300 - 0.6 volt 



After switching into the active region, A — —50, r„ || Ri, = 50 K, and the reflected 
source impedance is 1 K. From Eq. (6-20) 



£,(0+) 



ii + 5W^ 300 = - 2 - 34volt8 



which is well above the cutoff value of —3 volts. The total grid change is —2.94 
volts, and the plate drops by the same amount. 
From Eq. (6-24), 

e b (t) = 297 - 0.6 X 10«i 

The plate runs down 97 volts to the 200 volts at which the diode conducts in 

97 



h - <T6 X 10 " 



During this interval, the grid rises by 



AE, = 



to 



Aff t 97 
A = 50 
e«(«i) = -0.4 volt 



162fisec 



1.94 volts 



6-6. Recovery-time Improvement. The long recovery time is pri- 
marily a function of the large plate load of the Miller-sweep tube. If 



Sec. 6-5] 



VACUUM-TUBE VOLTAGE SWEEPS 



187 



in the capacitor charge and discharge paths we could somehow replace 
this large source impedance by a much smaller value without otherwise 
affecting the circuit, then the recovery time would be reduced proportion- 
ately. One convenient method is to isolate the capacitor from the plate 
circuit by means of an intervening cathode follower (Fig. 6-13a). 

A cathode follower's gain, especially when using a high-jj tube and a 
large value of R K , is almost unity. Since AcfEi, = E b , the cathode 
follower couples everything happening at the plate of Ti to the capacitor 
through its very low output impedance (Fig. 6-13b). The over-all 
circuit is identical with those previously given in Figs. 6-7 to 6-9, except 




*1 ,, *2 "2 



^r 



I — WV — « |( » vw- 



-=•£« 



A CF E b 



R*< 



(a) (b) 

Fig. 6-13. (a) Miller sweep with a cathode follower included for fast recovery; (b) 
equivalent grid circuit of the sweep tube. 

that the source impedance must be changed from Rl or R L || r vt as the 
case may be, to 

i — v kk = — -7-t 

M2 + 1 M2 + 1 

R s2 is quite small: in the normal triode it would lie between 200 and 500 
ohms. If we compare this small resistance with the 10- to 100-K source 
impedance that was formerly present, we can see that the bottoming and 
recovery time constants (n and r z ) will be greatly reduced. Conditions 
during the circuit-voltage jumps, rundown, and recovery remain as 
before, except for the introduction of i? s2 in all pertinent equations. 

The isolation of C from the plate of the Miller-sweep tube allows 
extremely rapid recovery of the plate, limited only by any stray circuit 
capacity from point b of the switch to ground. It is completely inde- 
pendent of the new Miller grid recovery time constant of [from Eq. 
(6-29)] 

r ??— ^ (6-31) 



A 



-('- + 5 ! £l) c 



188 



TIMING 



[Chap. 6 



Assuming R, 2 = 500 ohms in the circuit of Fig. 6-13, this time constant 
is reduced from 50 to 0.9 /*sec. The new waveshapes are sketched in 
Fig. 6-14. Note that the rundown time constant [Eq. (6-21)] remains 
relatively unaffected by the circuit change. But bottoming occurs 
slightly faster, once the grid reaches zero, because of the reduction of t 2) 
which was previously given by Eqs. (6-27). Just before opening the plate 
switch, the charge on C was approximately #&,„»„; just after the switch is 
opened the equivalent generator of the cathode follower jumps to E. 




JE c (0-) 




Fig. 6-14. Miller-sweep waveshapes when a cathode follower is used for fast recovery. 

The change of circuit voltage divides proportionately across r c i and 
Tpn/iia + 1), with the new grid voltage becoming 



■Ee.max = 



Td 



r.i + r p2 /(M2 + 1) 



(E — -Ei, min ) 



(6-32) 



Equation (6-32) indicates that upon recovery a very large positive grid 
voltage pulse will appear. In an actual circuit, the peak will always 
be appreciably less than the value calculated above. Both the grid 
resistance, which decreases markedly at higher values of grid voltage, 
and any stray capacity present act to limit the maximum grid jump to 
10 to 20 volts rather than the 50 or 100 volts calculated. This portion 
of the cycle appears after the region of major interest (the linear run- 
down), and therefore we can accept a very gross approximation for the 
solution. It still explains, as well as necessary, the waveshape that will be 
seen on an oscilloscope. 

The cathode follower also performs an important secondary function 
by making the sweep voltage available at a low impedance level for 



Sec. 6-6] 



VACUUM-TUBE VOLTAGE SWEEPS 



189 



coupling to other circuits. We can tolerate a much greater degree of 
external loading at the cathode, without affecting the sweep, than could 
possibly be applied at the plate of the sweep tube. 

6-6. Sweep Switching Problems. The final sweep essential is the 
establishment of the capacitor discharge path which terminates each 
sweep cycle. Looking backward to Chap. 5, this function was served by 
the gas tube. However, any other bistate device, such as the controlled 
gates of Chaps. 3 and 4, might serve equally well. These circuits require 
an external control signal which opens or closes the gate at the proper 
point of the cycle and for a given time interval. But in reality the 
extra signal involves slight, if any, additional complexity. In order to 
stabilize the gas-tube sweep, we were also forced to inject an external 
voltage, the sync signal. Thus the gate control voltage might simul- 
taneously perform two functions, switching and synchronization. 




Fig. 6-15. Diode-bridge sweep discharge circuit. 

Suppose we concentrate our attention on only one of the possible 
switches, the diode bridge of Sec. 3-6, and apply it to the three sweeps 
considered so far in this chapter. Each has its own special capacitor 
discharge or charge problems, and in the aggregate they adequately repre- 
sent what might be expected in almost any switching situation. 

First consider the current feedback sweep of Sec. 6-2 (Fig. 6-1). The 
diode bridge, applied across the capacity (Fig. 6-15), should normally 
be open: all diodes should be back-biased (e„ negative). Unless the 
diode's reverse resistance is very large, the shunting of C by r r will 
adversely affect both sweep time and linearity. In addition, the diode 
capabilities limit the maximum sweep amplitude to somewhat less than 
twice the peak inverse voltage. The control pulse forces the bridge 
into conduction, shunting C with r t and thus discharging C. For com- 
plete discharge the control-pulse duration must be at least Ar f C. Pro- 
vided that the spacing between adjacent pulses is less than or equal 
to the maximum linear sweep time [Eq. (6-5)], the periodic discharge 
locks the sweep time to the control-pulse period. 



190 timing [Chap. 6 

The diode bridge may also be used in shunt with the capacitor of the 
bootstrap sweep of Sec. 6-3. Amplitude limitations, required control- 
pulse duration, and the mechanism of switching are identical with that 
of the current feedback sweep. However, in this sweep we can compen- 
sate for the shunting effect of r, which had previously increased the sweep 
nonlinearity. The input impedance R/{\ — A) is made negative and 
equal in value to r T by making the base amplifier gain slightly greater 
than unity. The parallel-resistance combination becomes infinite, ensur- 
ing constant-current charging of C, even in the presence of the diode 
bridge. 

In the triode Miller sweep (Sec. 6-4), the switching circuit and recovery 
path are completely independent. The bridge, inserted in series with 
the plate, simply replaces the switch of Fig. 6-7a. When it is pulsed 
to cutoff (diodes back-biased), the large reverse resistance r r in series 
with the plate greatly reduces the loop gain and effectively opens the 
plate circuit, allowing recovery. When the bridge again conducts, the 
next sweep cycle starts. An alternative is to shunt R L with a normally 
conducting bridge and by this means reduce the load resistance, and hence 
the gain, to zero. Once the pulse opens the gate, the change in resistance 
produces the initial grid drop, with the amplification, now permitted, 
starting the linear rundown. 

The conclusion which can be drawn from the above arguments is 
that almost any means of discharging the energy-storage element, inter- 
rupting the feedback path, or reducing the loop amplification to zero will 
satisfactorily control the sweep operation. Of course, we must ensure 
that the circuit enters its active region upon switching; if it remains 
saturated or cut off, the sweep may be delayed or may never even get 
started. 

6-7. Pentode Miller Sweep. Rather than separate the functions of 
gating and sweep into two independent circuits, they might well be 
incorporated in a single tube, a pentode. The suppressor is normally 
biased negatively enough to completely cut off the plate current. In 
order to turn the plate on and so complete the feedback loop, we must 
apply a positive control pulse at the suppressor. The subsequent circuit 
operation, the equal drop in plate and grid voltage, the linear charging 
and eventual bottoming, and finally the recovery are similar to those of the 
triode Miller sweep. Furthermore, with its greater amplification, the 
pentode improves the sweep linearity by a factor of 10 or more over that 
of the triode and also permits a greater sweep amplitude. 

The basic differences between the pentode Miller sweep of Fig. 6-16 
and the triode sweep appear in all boundary conditions dependent on the 
physical characteristics of the tube. Practically, this sweep would also 
include a cathode follower for fast recovery and a plate catching diode, 



Sec. 6-7] 



VACUUM-TUBE VOLTAGE SWEEPS 



191 



but in the interests of circuit simplicity, these elements have been 
omitted from Fig. 6-16. 

Under normal circuit operation, the initial conditions are the following. 
The control grid, returned through R to £», conducts and therefore will 
be slightly positive, just as in the triode. Since the suppressor cuts off 
the plate, the cathode, control grid, and screen grid act as the three 
elements of a triode. As a direct consequence of the positive control 
grid, heavy screen current flows, with the resultant voltage drop in R e 
lowering the screen voltage to 50 or 60 volts. 

Once the positive gating pulse raises the suppressor voltage to, or 
more usually slightly above, zero, plate current will begin flowing. In 
order to maintain the gating voltage substantially constant over the 
sweep cycle, the control pulse must be applied either through the very 
long time constant C,R, or directly from a d-c source. An argument 
similar to the one employed with 
the triode sweep leads to the same 
conclusion; upon switching on, the 
plate and control grid both drop 
by an amount not quite sufficient 
to cut off the tube. Calculation 
of the exact initial drop (about 10 
volts) is not as simple a process as 
previously described, because the 
screen characteristics now have the 
predominant control over the tube 
current (Sec. 6-9). The low grid 
voltage reduces the total pentode 
current almost to the vanishing point, and therefore the screen voltage 
simultaneously jumps to nearly Em. 

After the initial jumps, the grid starts rising toward £&& at a rate 
given by Eq. (6-23), beginning the corresponding plate rundown. The 
large time constant t\ is identical with that of the triode sweep [Eq. 
(6-22)]. The initial conditions, voltage jumps, linear rise, and recovery 
are illustrated in the sketch of the circuit waveshapes (Fig. 6-18), and 
for clarity we should periodically refer to them during the following 
discussion. 

The next question facing us is, Where does the sweep end? The higher 
pentode gain (A = 250) permits the plate to fall to about zero while the 
grid rises by only 1 volt, or even less, from its approximate initial value 
of — 10 volts. Consequently, the limits of sweep operation are no longer 
determined by the grid but depend instead on the plate's bottoming. 
Figure 6-17 shows a portion of pentode plate volt-ampere characteristics 
represented by a set of straight lines. The large value of load resistance 




t 2 1 
Fig. 6-16. Pentode Miller sweep. 



192 timing [Chap. 6 

Rl was, of course, chosen for the highest possible amplification, and 
therefore the load line will be almost horizontal. 

Note that no further voltage drop is possible after the plate falls to 
the point where the load line intersects the knee of the characteristics, 
Eb. mm- At this operating point the plate resistance changes to a very low 
value, about 10,000 ohms, and g m is reduced to zero. Since the circuit 
amplification also becomes zero, the Miller effect ceases. Plate bottom- 
ing occurs at 2 to 10 volts, depending upon the characteristics of the 
particular tube and the load resistance chosen. This value is so very 




Fig. 6-17. Miller-sweep tube — plate characteristics and load line. 



small in terms of the total sweep amplitude that it is often approximated 
as zero. A somewhat better approximation would be 



E b 



> = Rl Evb 



where r'„, the effective plate resistance, is the reciprocal of the slope of the 
«i.,mi» line. 

Since the grid voltage of the pentode remains almost constant dur- 
ing the linear plate rundown, the capacitor charging current is best 
represented by 7 C (0+). By substituting this current, instead of Eu/R, 
the necessary time for the complete plate rundown, which starts at 
-En, + E cl (0+) and ends at E b , mia , is 



RC 



E bb + E cl (0+) - E b „ 
-Em, - E cl (0+) 



RC 



AE b 
E^ - E cl (0+) 



(6-33) 



During this same interval, the grid rises by only AE b /\A\ volts, where 
AE b is the total change in the plate voltage. If the maximum plate 



Sec. 6-7] 



VACUUM-TUBE VOLTAGE SWEEPS 



193 



voltage is limited to an external control voltage E by a plate catching 
diode, the starting point of the rundown would be E + E cl (0+) instead 
of En + E c i(0 + ). As in the triode circuit, the sweep duration is a linear 
function of this control signal. 

In Eq. (6-33), the only terms dependent on the tube are E c i(0+) and 
Eb, a >n. These are directly proportional to Ebb, and therefore any small 
decrease or increase in the supply voltage would change both the numer- 
ator and denominator proportionately. Thus we conclude that both this 
sweep and the phantastron (Sec. 6-8) have excellent time stability with 
respect to power-supply variations. 




Ec max' 



Fig. 6-18. Pentode Miller-sweep waveshapes. 

Furthermore, the terms in Eq. (6-33) dependent on the tube, # c i(0+) 
and Eb. min , are both small compared with Em,. By making R L large, 
-Et.min is reduced almost to the vanishing point. The other term, E c i(0 + ), 
remains relatively constant over the life of the tube and does not change 
much from tube to tube. To a very good approximation we can say that 
this sweep is virtually independent of the tube. This is one of the major 
advantages negative feedback offers over the positive feedback employed 
in the bootstrap sweep. 

Even after the plate bottoms, the grid continues charging toward En, 
but very much faster, with a time constant reduced from r x = \A\RC to 

r£' = C(R + r') S CR 



194 timing [Chap. 6 

The grid, charging from only a few volts negative, reaches zero in a 
comparatively short time and begins conducting. All circuit conditions 
again change; the time constant becomes 

r4" = (r„ + r' v )G 

and the steady-state voltage changes back to its initial value E c i(0~). 

But the rapidly rising grid voltage increases the total cathode current, 
almost all of which now flows to the screen. Therefore the screen 
voltage will drop to slightly above its initial value, the small amount of 
current flowing to the plate accounting for the difference. 

Immediately upon the removal of the suppressor gating pulse, the 
plate cuts off and recovery proceeds as in the triode sweep, with all the 
voltage jumps and time constants found in a similar manner. The only 
additional consideration is that the positive grid jump also appears as an 
amplified drop at the screen. 

The pentode circuit, besides producing the linear plate rundown, 
simultaneously generates a large rectangular pulse of equal duration 
which has fast rise and fall times. This signal, which appears at the 
screen, is as important as the linear sweep. We might apply it to the 
normally cut-off grid of a cathode-ray tube, thus unblanking the oscillo- 
scope only during the linear sweep (supplied by the plate rundown). 
Since its period is well defined, it can be used in conjunction with a gate 
for accurate time selection. If differentiated, the negative output trigger, 
which is delayed from the initial positive trigger by the sweep period, 
serves either for timing measurements or to start subsequent operations. 

A few words are now in order concerning the special requirements 
we impose on the pentode used in this sweep. First, it should have sharp 
suppressor control over plate current flow to ensure definitive on-off 
circuit states. Secondly, since under plate-current cutoff conditions the 
total cathode current flows to the screen, the tube should have the 
capabilities necessary to dissipate the heat produced ; its maximum screen 
dissipation must be large. Only slightly less important is a small control- 
grid-screen-grid transconductance to ensure that the screen voltage will 
remain reasonably constant during the linear grid rise. Otherwise the 
screen degeneration reduces the effective plate-circuit amplification and 
thus increases the sweep nonlinearity. Special tubes, such as the 6AS6 
and the 6BH6, whose specifications satisfy the above requirements, have 
been developed primarily for Miller-sweep and phantastron applications. 

6-8. The Phantastron. For optimum circuit response, the ideal 
Miller-sweep gating pulse would be one having exactly the same duration 
as the linear plate rundown. The circuit would recover immediately 
upon the plate's bottoming, becoming ready, in the shortest possible time 
interval, to react to the next input pulse. But the Miller sweep itself 



Sec. 6-8] vacuum-tube voltage sweeps 195 

generates a pulse of just the proper duration, and therefore we might 
just as well let the circuit do its own gating. All that this requires is 
the coupling of the screen pulse directly into the suppressor. Since the 
two waveshapes, the sweep and gating pulse, are simultaneously pro- 
duced, we are not forced to cope with the problems of synchronization 
and phasing that we would be sure to meet in attempting to generate an 
independent control signal. 

Figure 6-19o shows the complete phantastron circuit including a 
cathode follower for fast recovery and a plate catching diode D\ for 
voltage control of sweep duration. We normally adjust the coupling 
network R c , R a , and R, to keep the suppressor at about —20 volts, i.e., 
sufficiently negative to ensure plate cutoff. And as in the Miller sweep, 
the screen conducts heavily, setting the quiescent voltage level at 50 or 
60 volts. 

Application of a narrow positive trigger at either the suppressor or 
the screen starts the sweep by raising the suppressor voltage to zero or 
even slightly higher, thus bringing the plate out of cutoff. The resultant 
drop in control-grid voltage almost cuts off the entire tube, and conse- 
quently the screen will jump toward Em. This jump, coupled through 
C c and the resistor network, keeps the suppressor turned on, even after 
the starting trigger disappears. C c serves to speed up the switching 
action by immediately coupling the sharp rise and fall of the screen 
voltage into the suppressor, thus counteracting the retarding effects of 
the stray circuit capacity. It functions in a manner similar to the capaci- 
tor in a compensated attenuator which is adjusted somewhat overcom- 
pensated; usually C e is very small, only 25 to 100 nnf. 

Diode Z>2 limits the maximum positive suppressor voltage to about 
5 volts, set by the voltage divider Ri and fl 2 . We pick this voltage to 
give the largest possible gain, since g m reaches its maximum value at a 
slightly positive suppressor voltage. In addition, the suppressor current 
increases with increasing voltage, and unless limited to a very low value, 
the power dissipation may exceed the tube's ratings. 

Until the plate bottoms, the sweep operation is identical with that of 
the Miller sweep of Sec. 6-7, taking the same time to generate the same 
waveshapes (Fig. 6-19). However, after the plate bottoms,- the drop 
in screen voltage is directly coupled to the suppressor and turns the plate 
off. The cathode follower allows fast plate recovery to its initial value 
E and also Contributes a large positive grid jump, which, as we expect, 
appears amplified at the screen. But this additional drop can only 
help to turn the suppressor off even faster. 

The grid-circuit recovery is extremely rapid, because of the cathode 
follower and the control-grid conduction path. And the system is now 
ready for the next input trigger. 



196 



TIMING 



[Chap. 6 



We may inject the positive trigger pulse at either the screen or sup- 
pressor, the speed-up capacitor effectively applying it at both elements 
simultaneously. The trigger not only starts the suppressor into the 
plate conduction region, but also aids the initial screen rise. Its ampli- 
tude and duration are not critical, provided, however, that it at least 
exceeds the threshold necessary for guaranteed switching. 




Trigger 



*C2 



e C 3 



£»,- 



£ c2 (o-) 



t-^c max 
. E cl (0-) 



AAi 1 ^ 


/ 


. 


_______^ 


h 


t 






<T> 








t 














t 






k ■ 






h. 



Fig. 6-19. Phantastron circuit and waveshapes. 



Example 6-2. The phantastron of Fig. 6-19 employs a 6AS6 with the plate load 
adjusted to give a gain of 250. We can assume that initially the grid drops to — 10 
volts and that the plate finally bottoms at 5 volts. Furthermore, in this circuit, 
Ebb = 300 volts and E = 200 volts. 

(o) What RC product is required to give a sweep time of 500 /isec? 

(6) Under these circumstances, what is the sweep nonlinearity? 

(c) Assuming that i? c i(0 + ) changes by 10 per cent, by what percentage would the 
sweep duration change? 



Sec. 6-8] vacuum-tube voltage sweeps 197 

Solution, (a) The total plate rundown is 

AE b = 200 - 10 - 5 = 185 volts 
From Eq. (6-33), 

RC = h £w, .j; 10 = 500 X 10-' X ~, = 825 X lO-'sec 

loo loO 

Let R = 1.65 megohms and C = 500 upd. 

(b) The change in the grid voltage over the complete linear rundown is only 

Thus the grid rises from — 10 to —9.26 volts out of a charging curve having a 300-volt 
maximum value. Substituting these limits into the linearity equation (5-22) yields 

NL= 12.5% X§^jS0.03% 

Actually, this value may be too small to have any significance. The nonlinearity of 
C over this voltage range and the various second-order effects would, at the very 
least, double the calculated value. 

(c) From Eq. (6-33), with E.i(0 + ) = -9 volts, 



The normal sweep is 



t\ = RC X 1 8 %09 
U = RC X ls Hio 
Thus, by dividing and expanding, 

t\ /l86\/310\ (185 + 1)(309 + 1) , 496 

= 1 + ,.„,,,. = 1.0087 



l l = (—\ (—\ - 
h ~ \185/ \309/ ~ 



h \185/ \309/ (185) (309) ~ (185) (309) 

The sweep time is increased by only 0.87 per cent when the initial grid drop is reduced 
by 10 per cent. 

Free-running Phantastron. The phantastron lends itself to self-trig- 
gering or free-running operation. By simply setting the quiescent sup- 
pressor bias within its base, the plate is normally conducting. Consider 
the circuit behavior immediately following plate bottoming. The screen 
voltage drop, together with the large sharp spike (contributed by the 
control grid's positive jump), momentarily drives the suppressor below 
plate cutoff. As the control grid recovers, both the screen and suppressor 
voltage follow it toward their quiescent values. But the suppressor 
eventually reaches a point where the plate can turn back on, and as a 
consequence, the switching cycle repeats. The screen, driven back 
toward En, pulls the suppressor up along with it. Rundown begins 
again, and the cycle keeps repeating. 

Synchronization of the phantastron is usually effected by converting 
the input signal to pulses, which are then used to turn the sweep on or 
maybe to turn it off. These control either the start or the sweep bottom- 



198 



TIMING 



[Chap. 6 



ing point, and the resultant regions of synchronization may be defined 
in exactly the manner of Sec. 5-6. 

6-9. Miller Sweep and Phantastron — Screen and Control-grid Volt- 
age Calculations. To attempt an exact solution for the pentode screen 
and control-grid voltages is to attempt an extremely difficult task. We 
would need a complete set of both plate and screen volt-ampere character- 
istics, and even then probably the best approach would be one of suc- 
cessive approximations, i.e., making a guess, checking it, and then making 
a more educated guess, until finally some guess agrees with the checked 
answer. 





Fig. 6-20. Pentode screen circuit models, (a) Plate cut off; (fe) plate conducting. 

However, we can quite simply find an approximate solution, by a 
method which still keeps the essential circuit behavior in the forefront. 
This is through treating the cathode, control grid, and screen as a triode. 
It follows that the screen model of Fig. 6-20a represents the circuit 
when the suppressor cuts off the plate and the total current flows to the 
screen. Rth and E T h are actually the Thevenin equivalents of the screen 
network. The screen parameters r cU and ju c2 are determined from the 
tube under triode- operating conditions, plate and screen connected 
together so that i c i = ik- 



Mc2 



de c i 



Al.i-0 



Tcii = 



did 



de c i 
dih 



Aei-0 



(6-34) 



The parameters of Eqs. (6-34) may be found directly from the manu- 
facturer's curves by adding the screen and plate current characteristics. 
For the two widely used sweep tubes, reasonable values are : 



Sweep tube 


liel 


r««, kilohms 


p 


6AS6 

6BH6 


25 
20 


5 

4.5 


3 
2 







Sec. 6-9] vacuum-tube voltage sweeps 199 

Using these values, and with e c \ = 0, the quiescent value of e c « may 
readily be calculated from Fig. 6-20a. 

tlTh T" ~cU 

Once the suppressor pulse allows plate current flow, then its sole 
role is to regulate the percentage of the total cathode current that flows 
in the plate circuit. But the total current remains predominantly a 
function of the screen and control-grid voltages, and therefore the form 
of the screen model is still consistent with the tube's physical behavior. 
In the positive suppressor region, the division of current between the 
plate and screen is almost constant and is independent of the suppressor 
voltage. This ratio may be expressed as 

p = A (6-36) 

The constant p depends on tube geometry and is also found from the 
manufacturer's curves; it may be taken as 3 for the 6AS6 and 2 for the 
6BH6. When p is 3, only one-quarter of the total cathode current flows 
to the screen. With the tube now operating as a pentode, 

ik = ib + id = (1 + p)»c2 

and ictp = — ; — r 

p + 1 

Substituting into Eqs. (6-34) yields 

r,i T = ^7— = (p + 1) -37- = (P + !)'«» 

OlcZp Oik 

Thus r c u must be multiplied by (p + 1) in order to account for the 
effects of plate current flow and the screen model will be changed to 
the one shown in Fig. 6-206. 

The equal drop in plate and control-grid voltage, upon switching, may 
be expressed solely in terms of the screen current by noting that 

E eJ (0+) £i -i b R L = -piciRL (6-37) 

where Rl is the effective plate load resistance. Substitution of Eq. 
(6-37) into the model of Fig. 6-206 establishes all the conditions necessary 
to solve for E c i(fl + ). And it follows directly that 

p , n+ \ pEthRl 

° lK ' HczpRl +(p + 1>c2« + R™ 



But since r c n(p + 1) + Rth <K hcipRl, 



Eth 



E cl (0+) S* - — (6-38) 

Mc2 



200 timing [Chap. 6 

Equation (6-38) is just the cutoff voltage of the screen circuit, treated as a 
triode, which verifies our assumption that the tube is driven almost to 
cutoff. At the same time the screen rises to nearly Eth, as might be 
found from Fig. 6-206. 

During the control-grid rise, the screen voltage droops slightly because 
of the amplification from control grid to screen, 

AcUi = ~ (p + l)rl"+ R Th (6 - 39) 

In order to prevent excessive changes in the screen voltage, the ampli- 
fication must be kept small. To this end we usually choose R Th to be 
of the same order of magnitude as r c2 «(p + 1). Therefore the gain is only 
10 or 12. Since the control-grid voltage changes by a fraction of a volt 
over the whole sweep cycle, the screen voltage will remain substantially 
constant. 

Example 6-3. We shall now consider the design of the screen-suppressor coupling 
circuit for the phantastron of Example 6-2 (Fig. 6-19). The desired quiescent condi- 
tions are E ci = 50 volts and E c % — — 20 volts. The two supplies at our disposal are 
+300 and -200 volts. 

?+300 



ScreenI 5Q 



Suppressor* -20 

-200 
Fro. 6-21. Model for Example 6-3. 

Solution. The model holding under plate cutoff conditions is shown in Fig. 6-21. 
We arbitrarily choose a 1-ma bleeder current through R a and R,. If this current is too 
large, appreciable power would be wasted; if too small, the resistors would become 
excessively large. Therefore 

70volts = 70K 

1 ma 
l^volts = lg0K 

I ma 

Since r c% , = 5 K, the screen current at 50 volts is 10 ma. Thus 11 nia must flow 
through Rt, and 

Be . 25p_, 23 K 

II ma 



VACUUM-TUBE VOLTAGE SWEEPS 201 

Prom these values, we can find the Thevenin equivalent of the screen circuit. It 
is a 278-volt source having an internal impedance of 22 K and 250 K in parallel, or, 
to a good approximation, 20 K. The grid voltage after switching becomes, from 
Eq. (6-38), 

#.i(0 + ) = - 21 %t S -11 volts 

The grid-to-screen amplification is [Eq. (6-39)] 

. _ 25 X 20 K _ 10 g 

A cUi (3 + 1)5 K + 20 K 

In the previous example the control-grid voltage only changed by 0.74 volt. Conse- 
quently the screen voltage will run down by only 9 volts from its starting point 
of approximately 278 volts. These values check quite closely with laboratory 
measurements. 



PROBLEMS 

6-1. (a) Prove that varying E in the circuit of Fig. 6-1 is a very unsatisfactory 
method of adjusting the sweep time. Consider the sweep linearity as E charges from 
10 to 100 volts. The final voltage remains constant at Eu.mai- 

(6) Under the conditions of part a, how will the sweep duration change if the plate 
is always allowed to bottom before the capacitor is discharged? 

6-2. The sweep of Fig. 6-1 uses a 0.05-mI capacitor for timing. Once the switch is 
opened, it remains open. Sketch the plate waveshape if the grid resistance in the 
positive grid region is 1 K. The model which should be used for the plate bottoming 
region is a resistor of r„ = 700 ohms (no controlled source) from the plate to the 
cathode. This model holds for e c > e& (Prob. 4-4). 

6-3. Assume that we desire to construct a bootstrap sweep but that we are unable 
to obtain an amplifier with a sufficiently stable gain A. Instead, we can make ft a 
function of A, so that, within rather narrow limits, the change in ft will compensate 
for the change in A and thus maintain the sweep time invariant. Find the required 
functional relationship, that is, ft = f(A). Find the approximate relationship when 
A is close to unity. Repeat if R = f(En). 

6-4. The only tube available for the sweep of Fig. 6-1 has y. => 5 and r„ = 5 K. 
The other circuit values are En = 300 volts, E — 100 volts, ft* = 50 K, and C =- 
0.002 4. 

(a) Sketch the plate waveshape, giving all values if the switch is opened at t = 
and closed when e„* reaches zero. 

(b) Calculate the NL of this sweep. 

(Hint: All the equations in Sec. 6-2 may not hold with a low-^ tube.) 
6-5. (a) Draw the capacitor and output waveshape on the same axis if A = 0.95, 
E - 100 volts, ft = 1 megohm, and C = 100 «rf in the circuit of Fig. 6-3. The 
capacitor is discharged when its terminal voltage reaches 100 volts. 

(b) Calculate the sweep NL. 

(c) Compare the results of part a with the results obtained when A — 1. For this 
comparison the capacitor is adjusted to maintain the same sweep period. 

(d) This sweep is adjusted by varying E. Plot the time duration versus K if 
E = 1/KE, (A = 1). 

6-6. The bootstrap of Fig. 6-22 employs a cathode follower as its base amplifier, 
(a) Sketch the voltage waveshape appearing at the cathode if the switch is opened 
at t = and closed once the grid-to-cathode voltage reaches zero. Label all voltage. 



202 timing [Chap. 6 

values, time constants, and times. (Hint: Replace the tube by the equivalent circuit 
seen when looking into the cathode.) 

(6) Show that within its active region this circuit may be represented by the model 
of Fig. 6-4. Specify the parameters. 

(c) Calculate and compare the sweep amplitude and linearity when the battery in 
series with the charging resistor is present and when it is absent. 



+200 v 



M -100 
r„-100K 




f+300v 

1 r„=50K 

/ il^f r c =2K 




Fio. 6-22 



Fig. 6-23 



6-7. The circuit of Fig. 6-23 makes use of the techniques discussed in Sec. 6-2 to 
generate a specific nonlinear sweep. Sketch ei, e*, and e 3 to scale if both switches are 
opened at t = and closed when the voltage across the tube, ebt, drops to 200 volts. 
What function does this circuit generate? Make all reasonable approximations in 
your solution. Justify any assumptions made. 

6-8. In the Miller sweep of Fig. 6-7, the switch is closed at t = and opened a 
short time after the plate falls to its lowest value. Sketch the plate and grid wave- 
shapes, giving the values of all voltages, times, and time constants. Compute the 
sweep nonlinearity of the linear plate rundown. The tube and circuit parameters are 



Rl = 240 K 
R = 1 megohm 
C = 1,000 «if 
n = 100 



r p = 50 K 
r» = 2K 
En = 300 volts 



6-9. Repeat Prob. 6-8 if the plate is returned through a plate catching diode to 
+200 volts. Pay particular attention to the time required for the linear plate run- 
down and for the time required for the complete plate recovery. 

6-10. (a) Compare the sweep linearity of the triode Miller sweep under the follow- 
ing conditions: 

1. The charging resistor is returned to 2?t*. 

2. The charging resistor is returned to a voltage equal to O.lEu,. Express the 
answer as a ratio (assume \A\ 5J> 1). 

(b) Prove that the sweep period is a linear function of the control voltage E. 

(c) Prove that the sweep period varies inversely with the voltage to which the grid 
resistor is returned. 

6-11. Figure 6-24 represents a variation in switching the Miller sweep on and off. 
Sketch the grid and plate waveshapes, labeling all times and break voltages. (Hint: 
Be careful in evaluating the initial conditions of each region.) The switch is opened 
at t = and closed soon after D t conducts. What is the largest voltage which can 



VACUUM-TUBE VOLTAGE SWEEPS 



203 



be used to back-bias the tube and still allow the sweep to start as soon as the switch 
is opened? 

9+300 v 



I100K 



2M 

500 w* 

)Y 



-w- 



D 2 



€) 



-100 v 



M =50 

r p =25K 

r c =500 



Fig. 6-24 



6-12. (a) Compute the initial grid drop, the time required for the linear plate run- 
down, and the recovery time in the improved sweep of Fig. 6-13a. The circuit com- 
ponents are 

Eu = 300 volts R = 1 megohm in = 20 

E = 0.75£» R k = 40 K r r! = 10 K 

Rl = 200 K m = 100 

C = 1 Mf r pl = 100 K 

Assume that the total stray capacity from the plate of 2\ to ground is 50 «if- 

(b) Sketch the waveshape at the cathode of the cathode follower and calculate the 
internal impedance at this point. 

(c) What is the maximum possible value of E before the grid of the cathode follower 
is forced into conduction? If E exceeds this voltage, what happens to the circuit 
response? 

6-13. The sweep of Fig. 6-25 is placed in operation by opening the switch <S at 
t = 0. It is closed again at t = 200 jusec. Plot the grid and plate voltage to scale 
from before t = until the circuit completely recovers. 

g m = 1,000 /jmhos 
/j-100 

c- 1,000 mi\ 



200v-= 




=-100v 



Fig. 6-25 

6-14. Show three methods of switching the triode Miller sweep which are adaptable 
to diode or triode gating circuits. Discuss any limitations on or modifications in the 
basic sweep behavior when each gate is inserted. Give the circuit of these gates 
together with their points of insertion and the gating signal requirements. 

6-15. In the phantastron circuit of Fig. 6-26, assume that the plate falls by 10 volts 
when the pulse is injected. The plate rundown ends when D, conducts. If the 



204 



TIMING 



[Chap. 6 



loop gain is assumed to be —200, sketch the plate and grid waveshapes. Label 
these plots with all voltage and time values. Calculate the sweep nonlinearity. 



9+300 




p=2.S 

r„~500 K 

g m - 1,000 /imhos 

r c2 =10K 

^ c2 =20 

r,,-lK 



Trigger 
D input 



4+20 A -100 
Fig. 6-26 

6-16. Plot the correct suppressor, screen, plate, and control-grid waveshapes for 
the circuit given in Fig. 6-26. Assume that screen degeneration results in a 40 per 
cent decrease in the control-grid to plate-circuit gain. Give all voltage values, times, 
and time constants. 

6-17. The circuit of Fig. 6-26 has been modified to the circuit of Fig. 6-19. Repeat 
the calculations of Prob. 6-15 if the additional circuit parameters are 



Triode n = 100 

r„ = 100 K 
R k = 200 K 



E = 200 volts 
Ri = 100 K 
R* = 5K 



Make all reasonable approximations in your calculations and assume that the stray 
capacity loading the pentode elements is completely negligible. 

6-18. A phantastron used as a linear sweep is shown in Fig. 6-27. As a first 
approximation assume: 

1. Cathode-follower gain ^ 1 

2. An initial drop of 10 volts at the plate 

3. r v = 525 K and g m = 2,000 jimhos during rundown 

Given the cathode voltage waveform as shown in Fig. 6-27, calculate the approxi- 
mate plate waveform for the first 300 iisec after a trigger is applied. At the end of 
this interval the tube is turned off. If the total capacitance from the phantastron 
plate to ground is 100 nrf, calculate the flyback time. 



p+300v 



200 v-= 




Fig. 6-27 



VACUUM-TUBE VOLTAGE SWEEPS 



205 



6-19. The phantastron circuit of Fig. 6-19 is adjusted so that the normal suppressor 
voltage is zero (free-running sweep). 

(a) Discuss the effects on the sweep waveshape of injecting a synchronizing signal 
into the grid, screen grid, suppressor, and plate. This signal consists of a pulse train 
of equally spaced positive and negative pulses with a spacing between adjacent pulses 
of 75 /isec. The phantastron free-running sweep period is 1,000 jusec. Which point 
would be the best place to synchronize the sweep? 

(b) Plot the regions of pulse synchroni- 
zation if the sync signal is applied as 9+300 
shown in Fig. 6-28. 

+300 



esync 





Fw. 6-28 

6-20. This problem is designed to investigate the region of free-running phantastron 
operation between the time that the plate bottoms and the time that the next sweep 
starts. We do not have to consider the sweep rundown, but can concentrate atten- 
tion on the screen and suppressor coupling. Suppose that a 6BH6 is used in this cir- 
cuit, biased at E e i = 100 volts and E cl = 40 volts when the plate circuit is opened 
and when Di is removed. With the plate supply used, the suppressor cutoff may be 
taken as — 15 volts. Assume that the positive grid jump drives the screen down to 
10 volts, from where it recovers with the grid time constant of 2 i«sec. The plate and 
grid are decoupled by a cathode follower. 

(a) Calculate the value of resistors in the bleeder network of Fig. 6-29 needed for 
proper biasing. 

(6) Sketch the screen and suppressor waveshapes, giving all voltage values and 
times. Assume that the plate starts at 300 volts and runs down (after the initial 
drop) to zero in 50 jusec. 

BIBLIOGRAPHY 



Briggs, B. H.: The Miller Integrator, Electronic Eng., vol. 20, pp. 243-247, August, 

1948; pp. 279-284, September, 1948; pp. 325-330, October, 1948. 
Chance, B.: Some Precision Circuit Techniques Used in Wave-form Generation and 

Measurement, Rev. Sci. Instr., vol. 17, p. 396, October, 1946. 
et al. : "Waveforms," Massachusetts Institute of Technology Radiation 

Laboratory Series, vol. 19, McGraw-Hill Book Company, Inc., New York, 1949. 
Close, R. N., and M. T. Kibenbaum: Design of Phantastron Time Delay Circuits, 

Electronics, vol. 21, no. 4, pp. 100-107, 1948. 
Puckle, O. S.: "Time Bases," 2d ed., John Wiley & Sons, Inc., New York, 1951. 
Williams, F. C, and N. F. Moody: Ranging Circuits, Linear Time Base Generators 

and Associated Circuits, /. IEE (London), pt. IIIA, vol. 93, no. 7, pp. 1188-1198, 

1946. 



CHAPTER 7 

LINEAR TRANSISTOR VOLTAGE SWEEPS 



We rightly expect that almost all the basic sweeps of Chap. 6 may be 
adapted for transistor operation. Of course, the reverse transmission 
path present within the transistor and the vast difference in impedance 
levels do not permit its automatic substitution for the vacuum tube. 
But if the required minor circuit modifications are made, a transistor 
will perform at least as well as a triode in many of the voltage sweeps, 
and even somewhat better in some of them. 

Throughout the following discussion the reader should refer back 
to the appropriate sections of Chap. 6, both to review the basic concepts 
of the active-element sweeps and as a means of recognizing the differences 
between the transistor and vacuum-tube circuits. Even though the 
fundamental defining equations may be the same, it is these very differ- 
ences which account for the proper operation of each sweep. However, 
the basic similarity that does in fact exist broadens our outlook by 
enabling us to separate the system's behavior from the individually 
chosen components. If the same circuit operates with either a triode 
or a transistor, what is to prevent it from also working when some other 
active element is substituted? 

7-1. Constant-charging-current Voltage Sweep. One of the simplest 
possible voltage sweeps makes use of the current source of the transistor 
for the linear charging of the sweep capacitor (Fig. 7-1). This circuit's 
operation is analogous to that discussed in Sec. 6-2, but as we note, the 
almost constant current output makes the introduction of additional 
current feedback unnecessary for many applications. Furthermore, since 
the collector current is controlled by the emitter input, a means of 
adjusting the charging rate is afforded at a terminal well removed from 
the sweep output. 

The approximate model given in Fig. 7-16 adequately represents 
the circuit behavior because the emitter and base resistances are so 
very small compared with the external controlling resistance R that 
their neglect will have no appreciable effect on the operation. From 
this model we see that the essentially constant emitter current is given by 

L = §-< (7-1) 

206 



Sec. 7-1] 



LINEAR TRANSISTOR VOLTAGE SWEEPS 



207 



where E cc is the emitter bias source. After the switch is opened, the 
collector charges from its initial value of E» toward the Thevenin steady- 
state voltage of 



Eut = —aleTe = — 



aE a 
R 



with the long time constant t\ = r c C. 

But in order to have a reasonably large value of current flow, R will 
have to be relatively small, i.e., no greater than several thousand ohms. 
Since r c is normally greater than 1.0 megohm, the output apparently 
charges toward a very large negative voltage. Once the collector drops 
to zero, the transistor saturates with the now conducting collector diode, 



nsntn 



~-Eu 



(a) 




VW — °— M- 



<fb 



-e- 



j- 






(b) 
Fig. 7-1. (o) ConstanfrKjurrent sweep and output waveshape; (6) active-region model. 

shorting the capacitor to ground. Only a very small percentage of the 
total exponential appears at the output, and therefore it seems reason- 
able to approximate this voltage by an absolutely linear rundown. 



e,(t) 



-E»-jjfUdt = Eu,-aj£t (7-2) 



Substitution of the bottoming value of zero into Eq. (7-2) yields a maxi- 
mum sweep duration of 



RC 



tE c , 



(7-3) 



Equation (7-3) points up the dependency of this sweep on the value of a 
and hence the necessity of recalibration upon the replacement of the 
individual transistor. 



208 



TIMING 



[Chap. 7 



Referring to Eq. (7-3), we see that since a low value of resistance was 
used in the emitter circuit in an effort to maintain linearity, we are 
forced to turn to relatively large capacity for any given sweep dura- 
tion. The use of high capacitor values is one of the characteristics 
of transistor sweep circuits that differentiates them from their vacuum- 
tube equivalents. 

If this sawtooth is intended to drive a following transistor stage, the 
relatively low input impedance to be expected may load the capacitor 
excessively, with a corresponding deterioration in sweep quality. 



Zf 



— Ecr 



==c 



c 2 



,,e. 



2 e. 



1 



n 



Fig. 7-2. Transistor switching of the sweep circuit. 

Switching in the circuit of Fig. 7-1 may be accomplished by shunting 
the sweep capacitor by the complementary p-n-p transistor shown in 
Fig. 7-2. Under normal conditions the additional transistor T 2 conducts 
heavily and shunts C with its low saturation resistance. Injection of a 
positive pulse through the input coupling capacitor C 2 raises the base 
voltage of T 2 above E bb and rapidly cuts off the switching transistor. 
During the presence of the pulse, the very high back impedance of T 2 
will not prevent the expected linear rundown. Of course, this switching 
pulse should be long enough to allow bottoming of the sweep. 

Example 7-1. The recovery of the sweep of Fig. 7-2 is effected by recharging C 
from an almost constant current source. This current must be of the opposite 
polarity to that used for the original linear charging, and it is supplied from the com- 
plementary p-n-p transistor. The amplitude of the recharge current, and hence the 
time required for recovery, depends directly on the size of Rl Consider a circuit 
where £w> = \E C c\ = 10 volts, R = 1 K, and ai = ai = 0.98 and where the recovery 



IK 



:=l10v 

+ 



Of (el 

o 



6, 



'« C 

— dfr 



?P2 



10v-=- 



J ' 

Fig. 7-3. Model of the circuit of Fig. 7-2 holding in the recovery region. 



Sec. 7-2] linear transistor voltage sweeps 209 

time must not exceed 5 per cent of the linear rundown. Under these conditions, 
what is the limiting size of iJj? 

Solution. To a good approximation, the circuit of Fig. 7-3 represents the behavior 
of the sweep immediately after Ti is switched back on. Both transistors are in their 
active regions, and C is fully charged. Moreover, in approximating this circuit's 
response, the small emitter input resistance is assumed to be zero and the shunting 
collector resistance is assumed to be too large to influence either the sweep or recovery 
times. During the linear, sweep, T* is off and the total current flowing into C is 

%,i = — ort.i = — a Yrr = ~ ».e ma 

During the discharge interval, the capacitor current can be found by writing the node 
equation at point A : 

i fl t — <xi,z — ai,i ™ ai 2 — 9.8 ma 
E u 10 



where 



(1 - a)Ri 0.02i£j 



Since the charge and discharge of C are always from constant-current sources, the 
ratio of times is inversely proportional to the current flow. Therefore, under the 
conditions of the problem, for the recovery time to be 5 per cent of the sweep time, 

»,! = 20i,i 
Or to satisfy this condition, 

t,« = 2li.i = 210 ma 

Substituting and solving for Ri yields 

B * ~ h R n g *\g = 2 - 375 ohms 

i\ (.1 — a)CJc<: 

Once C is completely discharged, current can no longer flow into it. To do so 
would forward-bias the collector of Ti, and this would shunt C with the two conduct- 
ing diodes of the transistor. 

7-2. Bootstrap Voltage Sweep. Bootstrapping entails positive feed- 
back of the sweep voltage around an amplifier having essentially unity 
voltage gain. By this method we endeavor to maintain a constant 
voltage drop across, and consequently a constant current flow through, 
the charging resistor. One transistor bootstrap, illustrated in Fig. 7-4, 
employs an emitter follower as its base amplifier. Over its complete 
active region, the output is almost exactly equal to the applied base 
voltage (the drop from the base to emitter is quite small). However, 
the small voltage difference that does exist is inadequate to ensure 
sufficient charging current flow, and therefore an additional battery must 
be inserted in series with the feedback resistor. 

Our starting point in the analysis of this emitter follower is the selection 
of the proper model for the transistor. By choosing one based on Fig. 
4-23, rather than a T model, the circuit is reduced to the two-node 
network of Fig. 7-46, where, in addition, the feedback network (R and E) 
was replaced by its Norton equivalent. Furthermore, the large collector 
resistance r d is essentially in parallel with the much smaller R,, and there- 
fore it may be neglected. 



210 



TIMING 



[Chap. 7 



Since the transistor is a current-controlled device, it is more informa- 
tive to consider any variation in the circuit current instead of the resultant 
change in voltage. The input controlling current i\ divides between the 
external base-emitter resistance R and the transistor internal input 
impedance r' u . The actual base input is 

R 



it 



;ii 



(7-4) 



r'n + R 

where r' n may also be expressed in terms of the T parameters as 

r' n = n + r„{\ + 0) 

By substituting Eq. (7-4) into the model, the controlled source may be 
written as /3'i'i instead of as /34. Here 

R 



£' = 



r' n + R 







(7-5) 



As a result, the circuit of Fig. 7-46 may be replaced by the simplified 
model shown in Fig. 7-5. 





(a) 



(b) 



Fig. 7-4. (o) Transistor bootstrap sweep circuit; (6) model holding within the active 
region. 

With the switch closed, the transistor must be in its active region. 
It cannot be cut off because the supply current I flowing through R and 
R, produces a voltage drop which forward-biases the emitter base diode. 
The current through R e in the active region is found from the model of 
Fig. 7-5: 

i a = (1 + /3')*'i - / 



But the drop across R || r'n must also equal the drop across R e 

EtQ-) = [(1 + P')ii(0-) - I]R. = -ii(0-)R || r' u 
Solving this equation yields 



*i(0-) = 



IRe 



R II r'n + (1 + /?')«« 



Hence 



(7-6) 



Sec. 7-2] 



LINEAR TBANSISTOB VOLTAGE SWEEPS 



211 



Once the switch is opened, the current that formerly bypassed C 
begins charging it. The circuit is not disturbed when the sweep starts, 
and no voltage jumps appear. Thus the initial value of the charging 
current is 



*,(0+) = / - ti(O-) = 

In general, |8'jR e S> R || r'„ and 
P 



i,(0+) S : 



7 I (7-7) 



R |[ rjx + PR, 

R II r' u + (1 + P)R. 



R\\r' u 



«'i., J_ (1 + P')h 



V 



I 

o 



fix 

■5- 



l S e 



t 
*2 



1+/3' 

We might now note that almost 
the complete bias current goes to 
charge the capacitor. The remain- 
ing small amount sustains the 
proper operating point of the 
transistor. 

Charging continues until the 
transistor saturates. But the point 
at which this occurs is known; it is when the drop across R, rises to E». 
The corresponding limit of ij. is found from 

E 2s = E a = [i u {\ + p) - I]R e 
I R e + Em 



Fig. 7-5. Simplified model of the tran- 
sistor bootstrap of Fig. 7-4. 



lu 



(1 + &)R. 



Referring back to the input node, the final value of charging current in 
the linear portion of the sweep becomes 



f,(*i) = / 



%\. = 



P 



1 +p 



,1 



En 



(7-8) 



(1 + P)R, 

The second term in Eq. (7-8) is the change in current over the sweep 
interval, and for maximum linearity it must be small compared with the 
initial value. From Eqs. (7-7) and (7-8), 



PI 



p p >>_ p~ 

K ti e 



For the purposes of comparison, we shall set E - E M . By also sub- 
stituting the value of p given in Eq. (7-5), the necessary inequality 
reduces to 

PR. » r'„ + R (7-9) 

We conclude that R should be small and that a transistor with a large 
should be used if acceptable linearity is to be achieved. Usually R, 
limited solely by the current capabilities of the bias source, would be 



212 timing [Chap. 7 

on the same order of magnitude as r' u , i.e., about 200 to 1,000 ohms. 
With the /S's available of 60 to 100 or higher, the inequality of Eq. (7-9) 
is not difficult to satisfy with reasonable values of R t . 

Example 7-2. Suppose that /3 = 50, R = r' u = 1,000 ohms, R, = 2,000 ohms, and 
Et>b = E = 10 volts. Under these circumstances /3' = p/2 = 25 (from Eq. 7-5). 
The initial charging current, given by Eq. (7-7), is 

^ 0+) = 2m x wo = 9 - 62ma 

This is only 4 per cent less than /. At saturation the charging current is reduced to 

^= 9 - 62 -26(P00T = 9 - 43ma 

The total change of current is only 2 per cent, and linear charging of C appears to be 
an acceptable approximation. Actually, the sweep is exponential, with the current 
decreasing toward zero. 

From Eq. (5-22), the sweep nonlinearity may be expressed as 

NL = 12.5% 962 9 7 )2 9 ' 43 = 0.26% 

Various second-order effects which have been neglected in this discussion, such as the 
decrease in |3 as the transistor enters the saturation region, may even increase the NL 
by a factor of 2. 

Since the preceding argument proves that sweep nonlinearity is small, 
the input voltage may be approximated by the linear rise due to constant 
current charging of C. 

ei(0 =£ = jfct ( 7 - 10 ) 

and the time required to saturate the transistor is 

h=RC^ (7-11) 

B must be small for good linearity, and consequently we are again 
forced to turn to large values of C to establish the required RC product. 
To generate a 1.0-msec sweep with the circuit of Example 7-2, C would 
have to be slightly larger than 1 /if. It must be emphasized that the 
RC term in Eq. (7-11) is not the sweep time constant but only a product 
resulting from the analysis. 

The approximate waveshape produced at the emitter is sketched in 
Fig. 7-6. This point is isolated from the timing circuit and serves as a 
convenient low-impedance point from which to take the output. 

Any circuit, such as the one of Fig. 7-4, requiring an expensive, isolated 
power supply is quite unsatisfactory for general use. However, in this 
particular sweep it is possible to replace the battery by a charged capaci- 



Sec. 7-2] 



LINEAR TRANSISTOR VOLTAGE SWEEPS 



213 



tor (Ci of Fig. 7-7), and provided that we allow only a slight discharge 
of this energy source over any cycle, the basic mode of operation will be 
quite unaffected. Of course, some provision must be incorporated for 
automatic recharging. Examination of Fig. 7-7 will show how this is 
accomplished. 



IWi 




Its 






£ 2 «r) 



Fig. 7-6. Output voltage of the bootstrap 
sweep of Fig. 7-4. 



df 




Fig. 7-7. Bootstrap sweep containing a 
self-charging current source. 



Under the normal operating condition of this circuit, the switch 
across C is closed. This permits the energy-storage capacitor to charge 
through R, and the conducting diode to E^,. Upon opening the switch, 
the sweep begins. As the base starts rising, the emitter voltage follows 
it. Since Cx remains almost fully charged over the complete sweep 
cycle, the voltage at the bottom of the diode, point x, becomes 

e z = En + Ae c 

where the gain A is very close to unity. 

Hence the very slightest increase of the sweep voltage automatically 
back-biases the charging diode D x , and it may be removed from our con- 
sideration during the linear sweep interval. At the end of the sweep, the 
switch is closed, C discharges, and the diode again conducts, finally 
allowing Cx to recover toward En,. 

If we neglect the small base current which is also supplied from the 
charge stored in Cx, then during the sweep, charge is simply transferred 
from Cx to C. Since the total charge in the circuit must remain constant, 
it follows that 

AQ = CEu, = Cx AE (7-12) 

where AE is the drop in voltage across Cx over the complete sweep 
interval. For best linearity AE should be as small as possible, leading 
us to conclude that Ci must be very large compared with C. An adverse 
effect introduced by the large storage capacity is the long time required 
for circuit recovery, a time primarily determined by the time constant 
ftjCi. 



214 



TIMING 



[Chap. 7 



In this sweep the mechanism employed for switching might be iden- 
tical with that discussed in Sec. 7-1. Alternatively, any of the controlled 
gates would function equally well. The time constant and switching time 
of the discharge path established would have to be included in any 
calculation of the total recovery time of the circuit. 

7-3. Miller Sweep. The voltage Miller sweep (Fig. 7-8) depends for 
its proper functioning, as did the bootstrap, upon the conversion of the 
current response of the transistor into proportional voltage amplification, 
which is now used to multiply the feedback capacity. Before we can 





(a) (b) 

Fig. 7-8. (a) Transistor Miller sweep; (6) model holding within the active region. 

discuss the over-all sweep-circuit response we must evaluate the voltage 
gain. The input voltage may be expressed as 



ei = r n t b 



(7-13) 



where r' n = n + (1 + ff)r e . The change in collector voltage correspond- 
ing to this driving signal is simply 



e 2 = —fliiRi 



(7-14) 



Any loading of the output by the feedback capacity has been neglected. 
Solving Eqs. (7-13) and (7-14) simultaneously yields the voltage gain 



A = - —r- = -g m R 2 

Ml 



(7-15) 



Equation (7-15) might be interpreted as saying that the transistor 
has an effective g m of /3/rij. With = 100, r e = 10, and n = 200, the 
g m is 82.6 millimhos, which is so very much larger than can possibly be 
obtained from any triode that a reasonable gain is ensured, even when 
using very small load resistors. 

After the switch is opened and the circuit becomes active, Fig. 7-8b 
represents the complete equivalent model. In calculating the initial 
base voltage by the method developed with respect to the grid voltage 



Sec. 7-3] linear transistor voltage sweeps 

of the triode sweep of Sec. 6-4, we obtain 

£i(0+) 



R* I ~lk I 



215 



(7-16) 



1 - A 

where I is the capacitor charging current and where the approximation 
is the result of substituting the high gain given by Eq. (7-15). The 
equation indicates that there is a small positive jump due solely to the 
current flow through the reflected resistance (Fig. 7-9a). It follows 
that the reflected capacity C(l — A) will be initially uncharged. 
The charging current at t = 0+ is 

Eth — 2?i(0 + ) ^ Em 
Rth R 



7(0+) = 



(7-17) 



where £i(0+) « E Th , Rth = R \\ r' n , and E Th = r' n E hb /(R + r' n ). The 
change in base voltage over the complete charging interval must be small 





(b) 

Fig. 7-9. (a) Equivalent base input circuit — Miller sweep; (fe) collector equivalent 
circuit. 

compared with Eti, for maximum sweep linearity. Consequently, con- 
stant current charging may be assumed and 

Em, 



«i(0 



t'u En, , 

R ^ RC(1 



A) 



t 



(7-18) 



Since PR ^> r' u , the initial-jump is quite insignificant. 

From the collector model of Fig. 7-96, the output voltage may be 
expressed as 

e t (t) S En, + IRi + Aei(t) 



= Etb \ 1 ~RC t ) 



The time required for the collector to bottom at zero is 

h = RC 



(7-19) 



After bottoming, the base voltage continues rising toward a new steady- 
state voltage with a much faster time constant. We find these from 
the model of Fig. 7-86 by shunting the current source /3i b with a conducting 
diode. This also reduces the input resistance. 



216 



TIMING 



[Chap. 7 



When the switch is finally closed, the base immediately drops back 
to zero. This same change in voltage must be coupled by C to the 
collector and will drive it even further into saturation. Recovery 
of the collector toward zero is quite rapid, with the time constant pri- 
marily depending on the resistance of the now conducting collector- 
base diode. Above zero, the capacitor recharges toward Em with the 
time constant t* = CR2. The complete base and collector waveshapes 
are sketched in Fig. 7-10. 

7-4. Compound Transistors. Both voltage sweeps depended for their 
linearity improvement on the gain of the feedback amplifier: the boot- 
strap on maintaining a gain close to unity, and the Miller sweep on 
developing the largest possible negative gain. If the two appropriate 
equations are examined, we conclude that for optimum operation the 

t 

Bit 
A 



~T~ 1 1 ' * 

] t 

|^ _ }.__^__„ 

|0 it, f ? 

I 1 T 3 

J J 



Fig. 7-10. Transistor Miller-sweep waveshapes (ei and ei are not drawn to the same 
scale). 

transistor having an a closest to unity should be chosen (jS very large). 

In an effort to approach unity a, and at the same time to reduce the 
dependency of the circuit behavior on the individual transistor, Darling- 
ton proposed using a compound arrangement in place of a single tran- 
sistor. Figure 7-11 illustrates the suggested configuration: for our 
convenience in the later discussion the approximate current flow into and 
out of each transistor element is indicated. The arrows mark the actual 
direction of the current in the p-n-p transistors shown. 

The small base current of the primary transistor (Ti), i e i(l ~~ «i), 
is amplified by the correction transistor (T2) and added to the output of 



T, 



«2(l-«l)»,.l 



|(l-or 2 )(l-ori)t c i 



6 



Sec. 7-4] linear transistor voltage sweeps 217 

T\. Thus the composite collector current is composed of two terms : 

ie = id + id = «i»«i + (1 — ai)ati.i 

The first term is the normal current 

transmission through any transistor, m y, *i>.i 

and the second term represents a e ° 

small additional correction current 

flow from T2 that raises the over-all „ „ v "j~ 

output to a value much closer to the | , 

input driving current. We should A A 

note that as long as the individual V__y 

transistor a's are less than 1, the 

total output current will always be less 

than i,i. The two transistors, taken _ _ „ _ 

. i-i -i * IG - 7-11. Compound transistor cir- 

together, can be said to act as a single cu i t 
compound transistor, one having 

<*c = <*1 + (1 — «l)«2 

= 1 - (1 - ai)(l - a,) (7-20) 

For example, when a\ = 0.98 and a 2 = 0.97, the composite current gain 
is 0.9994. 

Additional transistors may also be incorporated for further correc- 
tion, each amplifying the base current of the previous transistor and 
adding it to the over-all output. They raise a c by multiplying the second 
term of Eq. (7-20) by additional factors of the form (1 — a,). But regard- 
less of the number used, a c will never quite reach unity. These addi- 
tional transistors have a progressively decreasing effect on a c , and there- 
fore the composite unit is usually composed of no more than two or three 
junction transistors. 

Stability of the composite element with respect to the individual a's 
is also much better than that of a single transistor. If only small varia- 
tions are considered, then 

5oT - 1 ~ " 2 sr, = 1 ~ a > (7 " 21) 

Using the figures given above, a 1.0 per cent change in ai would cause 
only a 0.03 per cent change in the over-all current gain a e . Equation 
(7-21) is not valid with respect to large variations of ai or a 2 , and in this 
case the effect on a e would have to be found by evaluating Eq. (7-20) over 
the expected range of ai and/or a^. 

The composite transistor is not an unmixed blessing; further examina- 
tion of the circuit of Fig. 7-1 exposes serious drawbacks which severely 
limit the possible applications. For example, the temperature-dependent 



218 timing [Chap. 7 

reverse collector current /,o of T% still flows unchanged at the base of the 
composite unit. Since the value of ft is very large, only a very small 
base current is needed to control the complete collector current flow. 
But lea may well be of the same order of magnitude, making the problem 
of temperature stabilization extremely difficult. 

Furthermore, it can be shown that the input impedance at the com- 
posite base may be approximated by 

r in .„ S [r.i(l + ft) + r bl ](l + ft) (7-22) 

As ft and ft are very large, the base input will no longer approximate 
the ideal short circuit which is desirable in a current-controlled device. 
But for just this reason, this circuit configuration is more convenient 
for use as a voltage amplifier; the loading of the external source decreases 
with the increase in ft (refer to the requirements given in Sees. 7-2 and 
7-3). If ever a higher impedance were needed, additional padding 
resistance would be inserted in series with the emitter. It would be 
multiplied by the product (1 + ft)(l + ft) as it is reflected into the 
base circuit. 

PROBLEMS 

7-1. Compare the sweep linearity if the transistor sweep of Fig. 7-1 is first used in a 
grounded base connection and then as a grounded emitter circuit. In both cases R$ is 
adjusted to make the sweep period 1.0 msec. The other circuit parameters are a = 
0.98, r c = 0.5 megohm, C = 0.1 4, E bb = 10 volts, and E cc = -10 volts. Specify 
the required values for Ri. 

7-2. (a) Plot the locus of operation (i c versus e„) of the sweep of Fig. 7-2 on the 
collector characteristics and evaluate the sweep and recovery times. The component 
values are ai = at = 0.99, r c = 1 megohm, C = 1 nf, R = 2 K, R 2 = 5 K, Ebb = 
10 volts, and E cc = -10 volts. 

(6) Modify this circuit so that a very short input pulse will periodically discharge 
the capacitor, with the sweep starting immediately upon the termination of the pulse. 
Sketch the new circuit. Specify the necessary value of Ri if the discharge must be 
complete within 10 /isec. 

(c) Compare the necessary power requirements for the circuits in parts o and 6. 
Which one offers the most efficient operation if the spacing between sweeps is O.lti, 
ti, 5ti? (ti is the sweep period.) 

7-3. The switching transistor in Prob. 7-2o presents a resistance of 25 ohms from 
the collector to the emitter when saturated. 

(o) How will this term affect the output waveshape and period? Write an expres- 
sion for the sweep interval in general terms, calling the conduction resistance r„ and 
then evaluate this equation. 

(6) Plot both the output waveshape and the locus of operation for the above circuit. 

7-4. We wish to evaluate the performance of the sweep of Fig. 7-12 to see whether 
it yields the same response as the circuit of Fig. 7-1 without having recourse to two 
power supplies. We have at our disposal a transistor having p = 50 and r„ = 
1 megohm, a 20-volt power supply, and a capacitor of 1,000 wit. 



LINEAR TRANSISTOR VOLTAGE SWEEPS 



219 



(a) What values of R. and Rb are needed to give a sweep amplitude of 15 volts and 
a sweep duration of 15 Msec? 

(6) What value of L is required so that the change in emitter current over the com- 
plete sweep period will be less than 0. 1 per cent? 

(c) Sketch e%, specifying all values. 



?+20v 




Fig. 7-12 
7-6. The bootstrap sweep of Fig. 7-4 has the following component values: 



n - 200 


E = 6 volts 


r„ = 1 megohm 


E bb = 12 volts 


r, = 20 


R. = 1,000 


= 50 


C = 0.04 4 


R = 800 





(a) Sketch the sweep output, labeling all voltages and times. 

(6) Evaluate the sweep nonlinearity. 

7-6. (a) Express the sweep nonlinearity of the circuit of Fig. 7-4o as a function 
of (3. 

(6) The circuit and parameters are En = E = 10 volts, »■'„ = 1,000, R, = 1,000, 
and R = 500ohms. Forwhat range of /Swill the NL fall between 0.5 and l.Opercent? 

7-7. Prove that the bootstrap sweep of Fig. 7-4 may be represented by the model 
of Fig. 7-13. Find the values of R n , A, and / for the circuit of Example 7-2. Com- 
pare the sweep duration and nonlinearity with that given in the text. 



Fig. 7-13 

7-8. (a) In the bootstrap sweep of Fig. 7-4, the bias source E is equal in value to 
Ebb, but it remains constant, whereas Ebb varies by + 20 per cent from its nominal " 
value of 10 volts. Calculate the change in sweep duration when r u «• 1,200, «■ 50, 
R = 1 K, R, = 1 K, and C = 0.05 4. 

(b) The sweep of part a is connected in the configuration of Fig. 7-7 with C\ = 
10 /if- Compare the sweep duration at the limits of Em with the answers found in 
part a. 



220 



TIMING 



[Chap. 7 



7-9. Figure 7-14 represents an alternative mode of switching the bootstrap sweep 
of Sec. 7-2. However, once the collector is opened, the conducting base-emitter 
diode presents a resistance of only 20 ohms, instead of the 1,000 ohms (r n ) seen with 
the switch closed. Plot the emitter voltage and the controlling current i'i to scale, 
after the switch is closed at t = 0. Pay particular attention to the circuit behavior 
as the switch closes. The other components are R = 500, R, = 2 K, /S = 100, 
C = 2 id, and E = E bb = 20 volts. 




Fig. 7-14 

7-10. The bootstrap circuit of Fig. 7-7 uses a transistor with the following parame- 
ters: r' u = 900, /S = 10. With this transistor and with R, = 5 K, R - 1.5 K, C = 
1 id, and Ey, = 20 volts, find 

(a) The waveform at the emitter when the switch S is opened at t = and closed 
when the voltage is equal to E bb . (Note that is relatively small, which invalidates 
some of the approximations made in Sec. 7-2.) Label all time constants and find the 
time when S is closed. 

(ft) The sweep nonlinearity. 

7-11. In the Miller sweep of Fig. 7-8 the switch is opened at t = and closed once 
the collector falls to 1 volt. The transistor employed has the identical characteristics 
of the one used in Prob. 7-5, and the other circuit values are Ri = 10 K, R = 5 K, 
C = 0.01 id, and E bb = 15 volts. 

(a) Sketch the collector waveshape, making all approximations given in Sec. 7-3. 
Calculate the sweep nonlinearity. 

(b) Repeat part a if R is reduced to 1,000 ohms and C increased so that the period 
remains the same as in part o. 

(c) If r, varies by a factor of 2 over the dynamic range of the transistor, how will 
this affect the sweep period? Give a qualitative answer. 

h20v 

r.-lO 
r 4 -100 
10 K-£ ,„ f £10K 0-50 



=-lv 



Fig. 7-15 

7-12. We shall use the circuit of Fig. 7-15 as an alternative to the Miller sweep of 
Fig. 7-8. The external emitter padding resistance minimizes the effects of the chang- 




LINEAB TRAN6ISTOB VOLTAGE SWEEPS 221 

ing n and r« during the collector rundown. Furthermore, to prevent the transistor 
from being driven completely into saturation, a bottoming diode is connected to the 
collector. For the circuit values given below, calculate the complete collector and 
base waveshapes, from t = until steady state is reached. 

7-13. Repeat Prob. 7-10 when the single transistor is replaced by the compound cir- 
cuit of Fig. 7-11. Each transistor has the identical parameters given in Prob. 7-10. 
Do the approximations of Sec. 7-2 now hold? How would a third transistor affect 
the sweep period? Discuss the effects of the increasing input impedance on the 
sweep waveshape. 

7-14. Prove that the maximum base input resistance of a composite transistor, 
composed of two identical units, must be less than the collector resistance r« of one. 
Show that this holds regardless of the size of the padding resistance inserted in series 
with the composite emitter. 

7-16. The composite transistor of Fig. 7-11 is composed of a high-power transistor 
Ti with its characteristics improved through the addition of T 2 . If these two tran- 
sistors have the parameters listed below, what is the equivalent base input impedance, 
emitter input impedance, base-to-collector current gain, and maximum power dissipa- 
tion of the composite unit? 

a, - 0.95 a: = 0.99 
n = 100 n = 400 

r. = 20 r. = 15 

P m - 5 watts P m = 200 mw 

7-16. Calculate the composite characteristics of the n-p-n and p-n-p transistor 
shown in Fig. 7-16 (ri„,„ n»,i>, «„, and e ). Assume that the two units are identical 




in all respects except the direction of current flow. Would this configuration be help- 
ful in linearizing any of the voltage or current sweeps discussed in this chapter? 
Explain how. 

BIBLIOGRAPHY 

Clarke, K. K., and M. V. Joyce: "Transistor Circuit Analysis," Addison-Wesley 

Publishing Company, Reading, Mass., in press. 
Darlington, S.: Patent No. 2,663,806 (assigned to Bell Telephone Laboratories). 
Nambiar, K. P. P., and A. R. Boothroyd: Junction Transistor Bootstrap Linear 

Sweep Circuits, Proc. IEE (London), pt. B, vol. 104, pp. 293-306, 1957. 



CHAPTER 8 
LINEAR CURRENT SWEEPS 



The electric deflection of a high-energy electron beam imposes a 
severe strain on the circuit designer in that it requires an excessively 
large sweep amplitude. For this reason magnetic deflection, which is 
produced by a linearly increasing current in the deflection coil, is used 
instead. The most common example is a television receiver where the 
raster is generated by the magnetic deflection of the beam: a single 
horizontal scan taking about 53 /isec and a vertical scan about 16 msec. 
Radar display scopes and some electromechanical systems employ 
similar deflection circuits. Furthermore, a coil excited with a linear 
current may be mechanically rotated, thus generating a spiral sweep. 

Our treatment of current sweeps in this chapter will closely parallel 
the development of the various voltage sweeps, with, however, an 
inductance replacing the capacitor as the basic timing element. We 
should expect our thinking to be influenced by the previous discussion 
and should feel free to adapt any existing circuit to our current need. 
For purposes of comparison and in order to gain perspective, the reader 
might occasionally refer back to the appropriate sections of Chaps. 6 
and 7. 

8-1. Basic Current Sweeps. If the ideal inductance did exist, as does 
the almost ideal capacitor, then the problem of producing a linearly 
increasing current would become trivial; simply switching a constant 
voltage across the coil would suffice. Because E = L di/dt, the current 
would immediately become 

i = \t (8-1) 

Unfortunately all coils have distributed winding resistance and inter- 
winding and stray capacity; a typical iron-core deflection coil of 50 mh 
may have a resistance of 70 ohms and an effective capacity of 200 ft/if- 
These parasitic elements are generally represented by the lumped param- 
eters (Ri, C) shown in the basic sweep circuit of Fig. 8-la, and their effects 
on the sweep waveshape must be accounted for in any analysis. 

Since we are interested in a linear sweep, the portion of the cycle 
devoted to the initial charging of C must be much smaller than the 

222 



Sec. 8-1] linear current sweeps 223 

time required for the coil current to build up to its final value. Immedi- 
ately upon closing the switch in Fig. 8-la, the complete circuit current 
flows into C. Thus the beginning of the sweep may be represented by 
the model of Fig. 8-16, which implicitly assumes that the start of the 



£.=. 





(b) 



R, 



E— 



4 \ \ 



e L gi 






(c) (d) 

Fig. 8-1. (a) Basic current sweep; (6) approximate model holding during initial 
charging; (c) model representing the current-sweep region; (d) recovery-region model. 

inductive-current build-up will be delayed until the terminal voltage 
reaches E. The time constant of this circuit, 



ti = R.C 



(8-2) 



is quite small, and the delay is proportional (h — 4ti) ; even when charg- 
ing through 500 ohms, the 200-niti capacity introduces a delay of less 
than 0.5 /usee (Fig. 8-2). 

Sweep Region. We can now assume that the circuit enters into its 
sweep region, which is defined by the model given in Fig. 8-lc. Neglect- 
ing the small current change occurring during the charging of C, the 
time response becomes 



it 



E 



R, + Rl 



(1 - e-"") tj = 



tli ~\~ R» 



(8-3) 



To justify our separating the response into these two segments, 
the sweep time constant ti must be much longer than r\. In the case 
considered above (L = 50 mh, R L — 70 ohms, and R, = 500 ohms), 
T2 = 88 usee. If R, is further reduced, the separation between n and r 2 
increases and the assumption made above becomes even more valid. 



224 timing [Chap. 8 

Moreover, in order to ensure good sweep linearity, the charging interval 
U must be limited to some small fraction of t*. 

Suppose that we employ this coil in a deflection circuit where the 
current must reach a peak of 200 ma in 53 Msec. From Eqs. (8-3), the 
necessary supply voltage is found to be 



E 



570I m 



= 252 volts 



The inductive component of the voltage decays to 138 volts over the 
sweep interval. An additional 14 volts (200 ma through 70 ohms) 
appears across the resistive component, raising the final coil voltage to 
152 volts. 




Fig. 8-2. Sweep current and voltage waveshapes for the basic sweep of Fig. 8-1 (switch 
in position 1). 

The waveshapes produced during this portion of the cycle are sketched 
in Fig. 8-2. Note the delay in starting the sweep and the compara- 
tively small voltage change across the complete coil and the large voltage 
change across the inductive component. 

Sweep Recovery. At the end of the required sweep interval, the switch 
is thrown to position 2. The response of the coil with its parasitic 
capacity and external damping, Rd, is found from the single node equa- 
tion of the circuit in Fig. 8-ld. We assume, in writing Eq. (8-4), that the 
series coil resistance will have but little influence during the portion 
of the recovery time with which we shall be concerned. 



% + h + ir = C 



+ if edt + w D = ° 



(8-4) 



The roots of Eq. (8-4) are 



Pi, 2 = 



1 



2RdO 



± V(^c) 2 - ic (8 " 5) 



Two bounding possibilities serve to delineate the response character- 
istics of the network. In one case. (l/2.RpC) 2 < 1/LC and the circuit 



Sec. 8-1] linear current sweeps 225 

exhibits a damped sinusoidal response. Here the poles are located at 

Pi,2 = — a + jfi 

1 



Where " = 2R D C f> = Vo>S-«* »<? = LC 

The complete solution of Eq. (8-4) may be expressed as 

4 = Ae-<" cos (fit + <t>) 

If the system is but slightly damped, a <K /S. By including the initial 
conditions, which are the final voltage and current reached during the 
linear sweep {E f and 7 m of Fig. 8-2), 

ih = I<n.e~ at cos wo< (8-6) 

e L ££ - Jjj I m e-"> sin wot (8-7) 

Since the damping factor a is so very small, the maximum coil voltage 
occurs when sin u4 — 1 (in 34 cycle of / or at t = w y/LC/2) : 

eL.„„ ^ - Jg In, (8-8) 

For the coil considered above, this peak reaches almost —3,200 volts 
only 4.98 jusec after recovery begins. Figure 8-3a illustrates the rapid 
oscillation and slow decay of the recovery portion of the sweep waveshape. 
A second bounding location of the roots of Eq. (8-4) occurs when 
Rd is adjusted to critically damp the response (Rd = \/£/4C). The two 
roots are now identical, both lying on the real axis at 

1 

Pl.2 = — 



\/LC 



When the initial conditions are included, the solutions for the critically 
damped case become 



%L 



ei = 



\L + VLC) 

(JL. + UY 



exp VW (8 " 9a) 

exp-^ (S-%) 



The peak voltage is much smaller, and it occurs sooner; for the same coil 
considered above, it reaches —2,380 volts at t = 3.16 Msec. Figure 8-3b 
shows this portion of the complete waveshape. As Rd is further reduced, 
the poles separate along the real axis. Even though the peak voltage is 



226 



TIMING 



[Chap. 8 



reduced below the value shown in Fig. 8-36, the increase in the time 
required for final recovery generally prevents the use of a very small 
damping resistance. 

In comparing Fig. 8-3o and b, we conclude that the fastest possible 
recovery corresponds to a single half cycle in the completely undamped 
case. For periodic sweeps the "switch" is returned to position 1 at the 
time that the coil current reaches its minimum point and the capacitor 
voltage returns to zero. The current will build up from approximately 
— /„ instead of from zero, and the sweep amplitudes and time will 
increase accordingly. 



-3,200 v 






T"*\J 


E r 


\ 3.16 /isec t 


2,380 V 


I 



(a) 



(b) 



Fig. 8-3. (a) Recovery waveshape when lightly damped; (6) recovery waveshape 
when critically damped. 

8-2. Switched Current Sweeps. Figure 8-4 illustrates the application 
of a transistor to the basic sweep of Fig. 8-1. We shall first assume that 
the circuit is initially inert; i.e., it contains no stored energy. When a 
pulse is injected, the transistor switches from cutoff to saturation, con- 
necting the full supply voltage across the inductive component of the 
circuit. (See the model of Fig. 8-5a.) Because the saturation resistance 
of the transistor r, is quite small, the delay in starting the sweep, due to 
the stray coil capacity, will be insignificant. Therefore, except during 
the oscillatory recovery interval, our discussion of this circuit will com- 
pletely neglect C. 

Immediately after switching, the operating point moves to the origin 
of the collector volt-ampere characteristics and the current starts increas- 
ing toward the intersection of the steady-state load line, —1/Rl, with 
the saturation resistance line of the transistor characteristics (Fig. 8-6). 



Sec. 8-2] linear current sweeps 227 

This steady-state current and the charging time constant may be found 
directly from the model of Fig. 8-5a. 



•/«»i — 



Em 



Rl + r, 



T\ 



Rl + t, 



(8-10) 



In general, Rl 3> r, and both terms of Eqs. (8-10) are primarily deter- 
mined by the coil and battery; they are almost completely independent 
of the transistor. 



L,R L ,C S 




Fig. 8-4. A switched current sweep — the transistor is driven from cutoff to saturation. 



-fSoooooj WV- 



-=-E u 




=rE u 



(a) 



r — nmum^- 



(b) 



-WA- 



T 



(c) 

Fig. 8-5. Models for the sweep of Fig. 8-4. (a) Charging region — transistor saturated ; 

(6) recovery region — transistor cut off; (c) recovery region — collector-base diode 
conducting. 

The collector current build-up is permitted to continue until it reaches 
I m , a value somewhat less than al-i, where Zi'is the peak injected emitter 
current. If the current build-up continues past this point, the transistor 
will enter its active region and the increase of the collector resistance 
(from r s to r c ) will radically distort the sweep waveshape. 

After the input pulse is removed, the transistor becomes back-biased. 
Since the only damping present is the very large reverse resistance of the 
transistor and the small coil resistance (Fig. 8-56) , the output waveshape 
is a damped oscillation similar to that shown in Fig. 8-3a. 



228 



TIMING 



[Chap. 8 



In this region the coil current and collector voltage follow the elliptical 
trajectory drawn on the collector characteristics of Fig. 8-6. At point A, 
the energy of the complete system, 



w L = y 2 Li n 



(8-11) 



is stored in the magnetic field of the coil. Along the path AB the stored 
energy is transferred to the electric field of the capacitor. As the coil 




Fig. 8-6. Trajectory of the collector voltage and coil current for the switched sweep 
of Fig. 8-4. The dashed portion is the path taken in the oscillatory recovery region; 
the solid-line segments, the path followed in the linear-sweep region. 

voltage reverses and builds up to the negative peak of E v , the coil current 
decreases to zero. At point B, 



W c = }iCE p * 



(8-12) 



From Eqs. (8-11) and (8-12), 



E. 



which is identical with the result given in Eq. (8-8). 

Because of the transistor's limitations, only a small voltage can be 
tolerated during retrace. By shunting the coil with additional resistance 
and capacity the peak can be reduced to an acceptable value, but this 
also reduces the frequency of oscillation and increases the portion of the 
sweep which must be allocated to recovery. If this time must be mini- 
mized, while still limiting the peak voltage, either smaller inductance 



Sec. 8-2] linear cukbent sweeps 229 

deflection coils can be used or two transistors may be placed in series, 
reducing the drop across each to one-half the total voltage present. 

Along the second portion of the elliptic trajectory BD, the energy is 
retransferred from C to the magnetic field of the deflection coil. The 
voltage now decreases, and the current builds up to — I m . The dashed 
portion is the path taken in the oscillatory recovery region and the solid- 
line segments, the path in the linear-sweep region. 

Slightly after one-half cycle of the oscillation (Figs. 8-6 and 8-7), 
the positive voltage developed across the coil exceeds the negative 
supply and the collector becomes forward-biased with respect to the base. 
A conduction path through this diode now exists irrespective of the 
emitter condition. The model holding changes from that of Fig. 8-56 
to the one of Fig. 8-5c. Current now builds up from the negative peak of 
— I m toward a new steady-state value 



J««2 = 

with a new time constant 

Ti = 



Eu 



RL + Tf 



R L + r, 



The charging path is now along the straight-line segment DO in Fig. 8-6. 
In order for the two portions of the sweep to have an equal slope and the 
same steady-state current, a transistor with r/ = r, must be selected. 
When R L is large, it will predominate and a much greater degree of 
unbalance in r/ and r, can be tolerated. 

To generate a periodic sweep, the emitter must be periodically switched 
from off to on. The time of the application of the input gating pulse 
is not extremely critical provided that it occurs before the coil current 
rises to zero. Under these conditions the transistor changes state 
smoothly, from its operation as a forward-conducting collector-base 
diode to its saturation region. Build-up continues along the path DO A 
to I m . Along this line segment the charging equation is 

hit) = I.. - (/.. + I m )e-» (8-13) 

And from Eq. (8-13), the time required for the current sweep to build up 
from — J„ to I m is 

<i - t In {" + { M (8-14) 

At the end of the linear-sweep interval the excitation is removed 
and the circuit recovers as discussed above. The portion of the period 



230 



TIMING 



[Chap. 8 



devoted to the flyback constitutes only one half cycle of the oscillatory 
wave, or 

1 IT 



U = 



2/o 



«0 



= tVlc 



(8-15) 



We have seen how a periodic excitation leads directly to a periodic 
sweep with half of the linear-sweep interval depending on the saturated 
transistor (external excitation) and the other half upon the self-excitation 
(stored energy) of the diode characteristics. 




Fig. 8-7. Periodic sweep produced by the circuit of Fig. 8-4 upon square-wave exci- 
tation. 

We should note that this sweep circuit is extremely efficient. During 
the half cycle that the transistor is saturated, energy is transferred from 
En to the coil. During the half cycle when the reverse current flows 
through the conducting collector-base diode, the stored energy is returned 
to the power supply. Of course, some power is dissipated over each 
half cycle in the circuit resistance. 

Satisfactory switching may be obtained from a square wave of current 
coupled into the emitter through an emitter follower. The sweep pro- 
duced is shown in Fig. 8-7. 

Example 8-1. We wish to see whether the sweep of Fig. 8-4 can be used in the 
horizontal-deflection circuit in a portable television receiver. Of the 63-jisec total 
period, the linear sweep requires 55 /isec, with 8 jisec allowed for the sweep recovery. 



Sec. 8-2] linear current sweeps 231 

The deflection yoke, which must be excited by a peak-to-peak current of 1.0 amp has 
the following parameters: L = 0.5 mh, R = 4.5 ohms, and C, — 50 pid. 

(a) With this coil, what supply voltage must be used? 

(b) How can the peak reverse voltage be minimized? 

(c) What is the sweep nonlinearity? 

Solution. Assume that both the saturation and the forward resistance of the 
transistor employed are 1 ohm. The time constant of the sweep interval is n = 
91 /jsec. Solving Eq. (8-14) with I m — 0.5 amp yields 

/„ = 1.7 amp 

Hence Eu, = 1.7(r, + Rl) = 9.35 volts. 

As C is increased, the peak reverse voltage decreases. The upper limit of C is that 
value that increases the half period of the recovery sinusoid to 8 /usee. From Eq. 
(8-15), 

c »=z6) 2=0013 ^ 

Thus the peak voltage becomes 



^ = - 5 VrHTp = 106volts 



Sweep nonlinearity is given by Eq. (5-20) : 

NL = 12.5% X 5 %i = 7.5% 

For acceptable linearity the coil inductance must be increased or the totel resistance 
decreased. 

Pentode Switched Sweeps. If a pentode is used as a switch (Fig. 8-8), 
because of the higher saturation resistance the start of the sweep will be 
slightly delayed. As the stray capacity charges, the operating point 
moves to the origin of the volt-ampere characteristic and the full supply 
voltage appears across the coil. This effectively places the tube in the 
region where it has a small value of r p , i.e., along the 2?&, miI , line, where 
r p drops to r' p . When a power pentode is used as the switching tube, 
r' p may be as low as 100 or 200 ohms. (The process described is similar 
to the bottoming of the pentode Miller sweep of Sec. 6-7.) 

The coil current increases exponentially from zero, with the time 
constant r = L/(r p + Rl), and as it does so, the operating point travels 
up the Et. miB line of the pentode characteristics toward the intersection 
with the load line. In order to maintain an almost linear build-up, the 
sweep will be terminated long before it reaches this steady-state current 
and before the tube becomes active. 

The removal of the control-grid signal turns the tube off. The large 
pulse now developed at the plate-coil current can be limited to a safe 
value by the addition of a suitable damping resistance. In the circuit 
of Fig. 8-8a, it is included in series with a diode, which is connected 
so that it will conduct only when the coil voltage reverses. 



232 timing [Chap. 8 

Since the tube cannot tolerate any reverse current flow, if we wish 
to adapt the pentode circuit for the most efficient recurrent operation, 
the coil must be shunted by a path which will support the sweep when 
%l < 0. In the transistor circuit of Fig. 8-4 this path was established 




L^^ 




(a) (b) 

Fiq. 8-8. Pentode switched sweeps, (a) Single-shot sweep circuit showing the con- 
nection of the damping diode; (6) periodic sweep showing the connection of the 
energy-recovery diode. 

through t9e collector-base diode; by analogy, an energy-recovery diode 
can be connected across the tube (Fig. 8-86). Just as in the transistor 
sweep, the diode will start to conduct slightly after the first half cycle 
of the damped sinusoid. At this time the coil voltage exceeds Em,, mak- 
ing the plate of the pentode negative with respect to ground. A com- 
parison of the vacuum-tube circuit of Fig. 8-8 
with the transistor sweep of Fig. 8-4 convinces 
us that the transistor is much more suitable 
for periodic current sweeps than the vacuum 
tube. The only advantage offered by the tube 
is that it can tolerate large reverse voltages 
during recovery. 

8-3. Current Sweep Linearization. The 
constant voltage charging of Sec. 8-1 is not a 
very satisfactory sweep for critical applications 
because of its delay in starting and its inherent 
nonlinearity. We now propose to reexamine 
the basic sweep of Fig. 8-106 and ask, What applied wave shape will 
ensure a linearly increasing current flow? 

In order to minimize the sweep delay, the initial portion of the excita- 
tion waveshape should be a voltage impulse. The impulse, shown in 
Fig. 8-9, is a pulse whose amplitude increases toward infinity as its 
duration decreases toward zero. Its area remains constant during the 



S(t) 



AT 



AT 

Fig. 8-9. Defining the unit 
impulse. 



Sec. 8-3] 



LINEAR CURRENT SWEEPS 



233 



limiting process. It follows that the unit impulse S(t) may be defined 
by the area integral 



CO* fAT 1 

jo W di =j AT dt = l 



(8-16) 



where, in the limit, AT = 0+ - 0. 

If we excite the coil through the source impedance R, with a voltage 
impulse of area K, then the stray capacity would be charged by the 
current 

. _ KS(t) 
% ~ R. 

From Eq. (8-16), the voltage across C is given by 

In order for this voltage to equal E at t = + , the weight of the impulse 
must be 

K = CR.E 

It is impossible to generate the ideal impulse, but it is possible to inject 
a finite-amplitude pulse at the institution of the sweep to speed the 
charging of C. Any large-amplitude pulse will reduce the sweep starting 
time by an appreciable factor. 







Impulse 






Wm 
















(a) 



(b) 



Fig. 8-10. (a) Required drive for the ideal current sweep; (ft) equivalent circuit of the 
sweep. 

To find the required drive voltage over the sweep region, we assume 
that the desired linearly increasing current is already flowing. 



i = kt = =? t 
fi 



(8-17) 



Substitution of this linear term into the circuit loop equation results in 



e,„ = L% + Rii = Lk + RJct 
at 



(8-18) 



234 



TIMING 



IChap. 8 



where Ri = R L + R,. Equation (8-18) defines the trapezoidal driving 
voltage shown in Fig. 8- 10a. The first term in the above equation 
{Lk = E) is the magnitude of the pedestal; i.e., it is the constant voltage 
which must be developed across an ideal inductance if the current is to 
increase linearly with time. The second term represents the linearly 
increasing voltage drop appearing across the resistive components of the 
circuit due to the current sweep produced in the coil. 

The individual components of the trapezoidal drive voltage necessary 
for a linear sweep may be generated separately, added together, and 
used to excite the coil. For example, by integrating a rectangular 
voltage wave in the input circuit of the emitter follower of Fig. 8-11, 
or alternatively in the grid circuit of a cathode follower, the voltage 
impressed across the coil would be of the proper trapezoidal form. The 




Fig. 8-11. Emitter-follower drive circuit for a current sweep. 

low source impedance of the driver stage limits the peak excursion of the 
output voltage to a reasonable value. A starting impulse must be 
separately injected. Many other modes of signal generation and coupling 
are also possible — through a transformer from a high-impedance source, 
capacitive coupling to eliminate the d-c component in the coil, or directly 
from a push-pull circuit. 

Example 8-2. The emitter follower used in the sweep of Fig. 8-11 employs as its 
active element the composite transistor of Sec. 7-4. Each unit has an a = 0.98. 
The 200-mh 30-ohm deflection coil must be excited by a linear-sweep current which 
reaches a maximum of 0.5 amp in 20 msec. In generating the input signal, the end 
of JBi is connected directly to the 50-volt supply. What must Ri, R s , and C be to 
satisfy the necessary sweep conditions? 

Solution. With the composite transistor, the large value of a e (0.9996) given by 
Eq. (7-20) means that the over-all gain is extremely close to unity. Consequently, 
the reflected input impedance of the emitter load (Rl and L) back into the base will 
be large enough so that its loading of the input RC network will be negligible. From 
Eq. (8-18), the height of the pedestal must be 



E - Lk - 200 X 10-« X 



0.5 

20 X 10" a 



5 volts 



Sec. 8-4] linear current sweeps 235 

The maximum height of the triangle, neglecting the small emitter resistance, is 

E, m = R t I m - 30 X 0.5 = 15 volts 

Assume that the initial current through the RC network is 1.0 ma. To obtain a 
5-voIt jump at the base input upon excitation, 

D 5 volts _ „ 
1 ma 

The remaining 45 volts must be developed across the other input resistor : 

= 45yolts _ 45 K 
1 ma 

The base must charge from 5 to 20 volts in the 20-msec sweep duration. From the 
exponential charging equation, 

,50-20 .„ . 
f i — t In -gf-- — =- ■" 49.5 msec 
5U — o 

Since Ri + R t = 50 K, C = 0.99 jif. We should note that the sweep duration is 
independent of the 6.7-msec coil time constant; in fact, the approximately linear 
sweep is now three times as long. 

8-4. A Transistor Bootstrap Sweep. For a second method of linear- 
izing the current sweep we might consider a circuit which is the dual 
of the RC voltage sweep of Fig. 6-2, i.e., the parallel combination of a 
coil and a conductance charged from a constant current source (Fig. 8-12). 
This circuit's node equation may be 
written as 



e £ =^-=-^ (8-19) 7 tQ 



Ml f 



At this point Eq. (8-19) should be 

j •j.i. i.\. i- .,, Fig. 8-12. Simple LG current sweep, 

compared with the equation written 

for the voltage sweep [Eq. (6-6)] and the argument employed with respect 

to that circuit reread. 

We know that bootstrapping generates an extremely linear voltage 

sweep. Since we now have at our disposal a current-controlled active 

circuit element, the transistor, we should consider the possibility of 

developing the dual circuit, a current bootstrap, as our current sweep. 

Choosing Eq. (8-19) for the starting point, we see that the replacement 

of /' by / + A c i L provides the correction term which is so essential. 

When this is done, 

eL = f + A * L - h (8-20) 

where A c is the current gain of the feedback amplifier. Figure 8-13a and 
b illustrate the current bootstrapping process defined in Eq. (8-20). It 
should be noted that the role of the current amplifier is analogous to that 



236 



TIMING 



[Chap. 8 



performed by the voltage amplifier in the voltage bootstrap. Both sup- 
ply the additional energy made necessary by the dissipative elements of 
the external circuit. By this means the system maintains either constant 
current or constant voltage charging of the energy-storage element. 
Since under these circumstances the dissipative energy remains constant 
(PR or EH3), the system must supply an ever-increasing amount of 
energy for storage in the capacitor or inductance. Unless an active cir- 
cuit element is present to cancel the effects of the circuit resistance (which 
limits the available storage energy), it is impossible to achieve an abso- 
lutely linear sweep. 



l+A c i L 




G | Qt /+A ^ 




E-IR 



(a) (b) (c) 

Fig. 8-13. Basic current bootstrap circuit, (a) Circuit connection; (6) schematic 
representation ; (c) TheVenin equivalent circuit. 

The role of the amplifier becomes somewhat clearer if we solve for the 
Thevenin equivalent of the current generator and its internal conductance 
in Fig. 8-136. One of the two voltage generators resulting is of the form 

e„ = RA c tL 

where R = 1/G. But since this voltage rise is proportional to the series 
current flow, it may be replaced by a negative resistance — RA C , as 
shown in Fig. 8-13c. The amplifier's action serves to reduce the over-all 
series resistance and to increase the circuit time constant. The current 
sweep is now defined by 



il = 



E 



Rl + R(l - Ac) 



(1 - e-«") 



R L + R(l - A e ) 



It is immediately apparent that there will be no improvement over the 
first circuit discussed unless A c is positive and greater than unity. When 
A c < 1 there will actually be deterioration of the sweep quality as a 
result of the decrease in the steady-state current and the time constant. 
Since both terms change in the same direction, the slope at the origin 
remains invariant with respect to any instability of A c . Any improve- 
ment in the sweep permits a larger amplitude output before the nonlinear- 
ity becomes excessive. 



Sec. 8-4] linear current sweeps 237 

Ideal constant voltage charging occurs when there is complete can- 
cellation of the resistance. From Fig. 8-13c we see that the necessary 
amplifier gain is 

A c = 1 + ^ (8-21) 

Usually R 3> Rl, and consequently the current gain required for linear 
charging will be but slightly greater than unity. Since we are dependent 
on the cancellation of two terms of equal magnitude for sweep lineariza- 
tion, any instability in i„ will have pronounced repercussions on the 
current waveshape, particularly on the time required to reach a predeter- 
mined final value [Eq. (6-10)]. Generally, the amplifier is adjusted so 
that the net resistance will be small but positive and so that any expected 
gain instability will never result in a net negative circuit resistance. The 
high degree of stabilization necessary, if the sweep time is to be very 
long, sets extremely stringent requirements on the base current amplifier. 
Previously, we proved that a trapezoidal voltage drive must appear 
across the physical inductance when a linear-sweep current flows through 
it [Eq. (8-18)]. Substituting the assumed ideal current u, = kt into the 
bootstrap equation (8-20) and equating the result to the known drive 
signal affords another method of determining the essential system 
conditions: 

I + (A c - l)kt 



«l = RiJd + Lk = 



G 



Comparing similar terms on both sides of the above equation yields two 
relationships : 

A e - 1 = R L G = ^ (8-22) 

I =LkG (8-23) 

The first of the above equations establishes the necessary system gain, 
and, as expected, it is identical with Eq. (8-21). The second condition 
[Eq. (8-23)] expresses the constant voltage charging of the inductive 
component of the external circuit. By rewriting this in the form 

Hi 
E = IR = L%= Lk 
at 

it also offers a convenient design criterion for the choice of the parallel 
resistance. 

The positive current gain of a single transistor is necessarily limited 
to somewhat less than a, and consequently a multiple-stage amplifier 
must be used in this bootstrap sweep. Figure 8-14 illustrates an appro- 
priate circuit, which employs current feedback both to increase the gain 



238 



TIMING 



[Chap. 8 



stability and as a means of adjusting the gain to the optimum value. 
This will generally be somewhat less than that given by Eq. (8-21), 
thus allowing a margin for any remaining amplifier instability. Since the 
coil resistance will not be completely canceled, if originally Bl was small, 
little or nothing will be gained by bootstrapping and the switched sweep 

of Fig. 8-4 might prove to be just 
as satisfactory. 

The current sweep discussed 
above is the dual of the bootstrap 
voltage sweep treated in Chaps. 6 
and 7. By analogy it would seem 
likely that a current-controlled 
Miller sweep, depending for its 
timing on the inductance, might 
also be practical. However, the 
impossibility of isolating the coil 
resistance causes any operation on 




Fig. 8-14. Base amplifier for a current 
bootstrap circuit. 



the inductance to be accompanied by an equal effect on the associated coil 
resistance. Multiplying or dividing both L and R L by the same factor 
leaves the time constant unchanged. Because there is no improvement, 
this method of linearization is never used. It is worth noting that not 
all forms of voltage-operated systems can, in fact, be converted to 
their duals. Often, as in this case, the unavailability of the needed ideal 
circuit element (pure inductance) will be the decisive factor. 





Fig. 8-15. Current sweep generated from a constant-current source. 

8-5. Constant-current-source Current Sweep. A contrasting line of 
thought actually obviates the generation of a current sweep. The tube 
or transistor is utilized solely as a voltage-to-current converter. A highly 
accurate voltage sweep is generated elsewhere and applied as a drive: 
in Fig. 8-15 the constant-current output of the transistor furnishes a 
driving current proportional to this voltage. We thus bypass the need 
to develop separately the trapezoidal voltage across the coil, the negative 
resistance in series with the coil, or any other correction term necessi- 
tated by the nonideal inductance. Moreover, by restricting the sweep 
generation to voltage waveshapes, we are able to employ an almost ideal 



Sec. 8-5] linear current sweeps 239 

energy-storage element (i.e., a capacitor), with consequent ease of 
linearization. 

The current flow through L, found from the model holding for the 
active region, is 

R ~ (8-24) 

where E d is the drop across the inductance. In general, Ri <5C r c and 
the small current component due to Eu, will be negligible; the coil current 
becomes a linear function of the driving voltage. 

At the end of the sweep cycle the input voltage drops to zero. The 
turnoff process is identical with the removal of the excitation in the 
switched sweeps of Sec. 8-2. Here also the recovery would be controlled 
either by an external damping diode and resistance or by a recovery 
diode for recurrent sweeps. 



PROBLEMS 

8-1. A coil having an inductance of 100 mh, a resistance of 20 ohms, and stray- 
capacity of 200 nid is charged through a source impedance of 1,000 ohms (Fig. 8-1). 
The sweep current must reach 150 ma in 50 fisee. 

(a) Specify the necessary charging voltage and calculate the sweep linearity. 

(6) Sketch the current and voltage waveshapes when the circuit is critically 
damped. 

(c) Repeat part 6 when the damping resistance is reduced to 1,000 ohms. Specify 
the peak voltage and the time at which it occurs. Make all reasonable approxima- 
tions. (Hint: Treat in the same way as the double-energy circuit of Chap. 1.) 

8-2. The coil of Prob. 8-1 is shunted with additional capacity so that one half cycle 
of the oscillatory recovery lasts for 60 iisec (fi d = «). Calculate the peak voltage 
and compare with the value found for the coil without additional shunting capacity. 
What does the additional capacity do to the initial sweep delay? 

8-3. The transistor used in the sweep of Fig. 8-4 is excited by a 1.0-msec-period cur- 
rent square wave. Both r. and r, at the collector are 5 ohms, r„ = 100 K, a = 0.97, 
Eu, = 20 volts, and the coil parameters are L = 0.05 henry, R = 10 ohms, and C = 
500 wf. 

(a) What is the minimum amplitude input that will ensure the transistor's remain- 
ing saturated over the full half of the sweep cycle? 

(6) Sketch the current and voltage waveshapes. 

(c) Calculate the sweep nonlinearity. 

8-4. As an alternative to the sweep of Fig. 8-4, the square-wave excitation is 
applied to the base as shown in Fig. 8-16. For the purposes of analysis assume that 
each equivalent diode in the transistor has a forward resistance of 5 ohms and a 
reverse resistance of 100 K. In addition, the collector-to-emitter saturation resist- 
ance is 10 ohms. The coil parameters are L = 100 mh, R = 20 ohms, and C = 
100 m&. (For this problem, remove the diode shunting the coil.) 

(a) Sketch the current if the 100-MSec square wave drives the transistor into 
saturation. 



240 



TIMING 



[Chap. 8 



(6) How might this circuit be modified to linearize the sweep? Explain your 
answer. 

9+30 v 




Fig. 8-16 

8-6. In the switched sweep of Fig. 8-8o the plate resistance of the pentode, when 
operating along the eb. m i n line, is 400 ohms. The coil used has an additional resistance 
of 100 ohms and an inductance of 500 mh. Once the current builds up to 100 ma, 
the tube is switched off (the plate voltage will not leave the e6,min line until this point). 
E bb = 300 volts. 

(o) Sketch the sweep current waveshape, specifying all times and time constants. 

(6) Evaluate the sweep linearity. 

(c) Repeat part a when the tube is not switched off until the current builds up to 
its maximum value. Assume that the plate resistance in the active region is 100 K. 
Calculate the peak plate voltage and give the time at which it occurs. Plot the 
operating locus on the plate volt-ampere characteristics. 

8-6. (a) If the coil of Prob. 8-5 has associated shunt capacity of 500 /i/xf, calculate 
the value of Rd necessary for critical damping. 

(6) Calculate the Rd that will result in a damping equation of the form 

i = ke~ at cos fit 
where a = fi. 

(c) Plot the voltage and current recovery waveshapes for parts o and 6 on the same 
graph. Which condition gives the best recovery response? 

8-7. Assume, for this problem, that the coil capacity in the switched sweep of 
Fig. 8-17 is zero. The plate resistance in the saturation region is r'„ = r p /(n + 1), 
and this region corresponds to £i., m in < -Ec.max. 

(a) Sketch the voltage and current waveshapes to scale. 
(6) Plot the locus of operation on the piecewise-linear tube characteristics, 
(c) Prove that the tube remains conducting when the pulse first disappears. 

y+300 v 
100 mh 



50 v 




100 • x ™ - 
A sec 4-20v- 

Fro. 8-17 



LINEAR CUEEENT SWEEPS 241 

8-8. A power-amplifier pentode having a saturation resistance of 200 ohms is used 
to excite a 50-mh 50-ohm coil in the periodic sweep of Fig. 8-86. The maximum 
saturation current of this tube, with zero control-grid voltage, is 250 ma. E» = 
400 volts. 

(a) Calculate the optimum value of iJi if the coil current must vary from — 100 ma 
to +100 ma over the sweep interval. 

(6)' Sketch the plate current, diode current, and coil current to scale. The oscilla- 
tory recovery time is limited by external capacity to 10 per cent of the linear-sweep 
interval. 

(c) Sketch the plate voltage waveshape. 

(d) Plot the plate characteristic trajectory of this sweep. 

(e) Calculate the sweep nonlinearity. 

8-9. The coil of Prob. 8-1 is driven by the output of an emitter follower as shown 
in Fig. 8-11. Assume that the output impedance of the approximately unity gain 
stage is only 10 ohms. The peak current of the single sweep cycle is 200 ma, and this 
circuit has a recovery diode and resistance critically damping the response. In addi- 
tion, shunting capacity is added to reduce the peak voltage across the coil to 100 volts. 

(a) If a 25-volt supply is used, what is the fastest possible sweep? 

(6) Calculate the necessary drive for a single sweep of 5 msec duration. 

(c) Sketch the voltage and current waveshapes to scale, paying particular attention 
to the recovery portion of the cycle. 

(d) Repeat part c when the diode is removed. 

8-10. The circuit of Fig. 8-16 is used as a periodic sweep. The transistor has 
a = 0.98, r, = 10 ohms, r b = 100 ohms, r d = 100 K, and the 10-mh coil being driven 
has a resistance of 10 ohms and 100 jujuf stray capacity. 

(a) What is the highest frequency sweep possible if the recovery is limited to 10 per 
cent of the sweep period? The peak-to-peak sweep current must be 5 amp and 
E kb = 30 volts. 

(h) Calculate the drive waveshape, at the base of the transistor, required to pro- 
duce a periodic sweep of one-half the frequency determined in part a. Ensure that 
the transistor will be back-biased during the recovery interval. Calculate the 
optimum value of R a . What function does this resistor perform? 

8-11. A power-amplifier triode is used as a cathode-follower driver with the deflec- 
tion coil inserted in the cathode circuit. For this tube /» = 10, r p = 1.5 K, and the 
plate supply is 400 volts. The same basic circuit is used for both horizontal and 
vertical deflection in a television receiver. Both deflection coils have L = 50 mh, 
R = 150 ohms, and C = 250 M/ if. For a single horizontal scan the current must 
increase to 200 ma in 55 //sec with 8 jusec allowed for recovery. The vertical scan 
requires a peak of 150 ma in 16 msec with 700 /isec left for the recovery. 

(a) Specify the necessary trapezoidal drive signal for the horizontal sweep. 

(6) Complete the design by calculating the diode resistance and recovery capacity. 

(c) Sketch the circuit waveshapes. 

(d) Repeat parts o, 6, and c for the vertical sweep. 

8-12. The coil of Prob. 8-11 is excited from a generator having 200 ohms source 
impedance. 

(a) Specify the drives necessary to produce a 200-ma peak 10-msec-duration 
periodic sweep of the following geometry: a parabola, i = KP; an isosceles 
triangle; a truncated isosceles triangle where the flat top is 3.3 msec long. 

(b) Sketch the waveshapes produced on the oscilloscope screen if the coil is rotated 
at 300 revolutions/sec and the sweeps are those given in part a. 

8-13. Prove that the current bootstrap circuit employing only a single transistor, 



242 



TIMING 



[Chap. 8 



such as the one shown in Fig. 8-18, produces a sweep of poorer quality than the sim- 
ple switched coil of Fig. 8-4. (a = 0.98, r, = 10 ohms, n = 100 ohms, and r c = 
1 megohm.) 



L 
i 



-A/W- 



it. R L , C 

r— TSW — I 




Rz 

-AAV- 



,1 

T 



Fia. 8-18 



8-14. In the base amplifier of Fig. 8-14 the two transistors are identical, each having 
= 50, r. = 10, n, = 200, and n = 100 K. The power supply Eu, - 30 volts, and 
R 1 = Ri = R t = 1 K. Plot the current gain as a function of R 3 . 

8-16. We wish to use the circuit of Fig. 8-19a to produce a perfectly linear current 
sweep i = kt after the switch is closed at t = 0. 

(o) For this condition, sketch the volt-ampere characteristic of N, labeling all 
slopes. 

(6) If i = kt with N properly adjusted, what is the value of kl 

(c) When N is represented by a two-stage current amplifier with feedback (Fig. 
8-196), specify the necessary current gain /S in terms of Ri, r„ and tl- 

(d) How does the time constant of the circuit vary as a function of /3, for small 
variations in 0, in the vicinity of the optimum value? 









N 



t "*! 



: R 0\fih r.l 



r B »R 1 



(a) 



(b) 



Fig. 8-19 



8-16. In the current sweep of Fig. 8-15 the driving voltage increases linearly from 
to 10 volts in 100 /isec. The circuit components are Ri = 100 ohms, L = 500 mh, 
Rl - 50 ohms, C = 0.01 i^f, Eu, = 20 volts, a = 0.98, r. = 20 ohms, and r c = 500 K. 

(a) Sketch the coil current and voltage to scale, specifying all values. Make all 
reasonable approximations. 

(6) Calculate the value of the damping resistance which should be included across 
the coil if the maximum collector voltage must be limited to 40 volts. Repeat part 
a, first when this resistor is directly across the coil, and then when it is in series with a 
damping diode. 

8-17. A series RL circuit is excited by a linear-sweep voltage from a low-impedance 
source such as a cathode follower. The total series resistance is 500 ohms, and the 
coil inductance is 1 henry. Evaluate and sketch the current and voltage waveshapes 
appearing across L 

(a) When t is long compared with the time duration of a single applied drive signal. 

(6) When t is short compared with the time duration of the drive signal. 



LINEAR CURRENT SWEEPS 



243 



(c) When the applied voltage has a peak amplitude of 100 volts and a duration of 
100 /usee. 

8-18. The triode current sweep of Fig. 8-20 is driven by a perfectly linear sweep 
voltage of 100 volts peak amplitude and 50 /tsec duration. The tube used is a 12AU7. 

(a) Sketch both the plate voltage and plate current waveshapes when the damping 
diode is omitted. (Make all reasonable approximations.) 

(b) Calculate the value of Rd necessary to limit the maximum plate-to-cathode 
voltage to 600 volts. 

(c) Repeat part a when the diode is included and when Rd is given by the answer 
of part 6. 



+300 




100 v 



8-19. (a) Prove that the coil current in the sweep of Fig. 8-15 may be expressed as 
i = Ae-'* + Bl +C 

where A, B, C, and t are constants of the circuit. 

(6) Evaluate the above constants in terms of the circuit and transistor parameters. 

(c) Is it possible to generate a perfectly linear sweep? What conditions must be 
satisfied if the sweep nonlinearity is to be minimized? 



BIBLIOGRAPHY 

Chance, B., et si.: "Waveforms," Massachusetts Institute of Technology Radiation 

Laboratory Series, vol. 19, McGraw-Hill Book Company, Inc., New York, 1949. 
Goodrich, H. C: A Transistorized Horizontal Deflection System, RCA Rev., vol. 18, 

pp. 293-321, September, 1957. 
Millman, J., and H. Taub: "Pulse and Digital Circuits," McGraw-Hill Book Com- 

pany, Inc., New York, 1956. 
Soller, T., et al.: "Cathode-ray Tube Displays," Massachusetts Institute of Teclu 

nology Radiation Laboratory Series, vol. 22, McGraw-Hill Book Company, Inc., 

New York, 1948. 
Sziklai, G. C, R. D. Lohman, and G. B. Herzog: A Study of Transistor Circuits for 

Television, Proc. IRE, vol. 41, no. 6, pp. 708-717, 1953. 



PART 3 

SWITCHING 



CHAPTER 9 

PLATE-GRID- AND COLLECTOR-BASE-COUPLED 
MULTIVIBRATORS 



9-1. Basic Multivibrator Considerations. Any closed-loop regener- 
ative system having two or more clearly defined stable or quasi-stable 
states, each of which is maintained without recourse to external forcing, 
might well be categorized as a multivibrator. The application of a low- 
energy-content stimulus, a trigger, starts the switching action which 
drives the operating point away from one stable state and toward the 
next. But the important attribute of the multivibrator which differ- 
entiates it from other multistate systems, such as a switch or relay, is 
that once the trigger brings the cir- 
cuit into its regenerative region, the e . o_. i A ^ e _ Ae . 

additional energy necessary to com- 
plete the transition is supplied by Sw 
the system itself. 

The necessary and sufficient condi 




e 



tions for any system to operate as a ^-A^e m ^ 

multivibrator may best be estab- „ „ , „ _. ' 

fished by considering the functional ^ ^ BaS1 ° re « eneratlve ^^ 

form of the multivibrator, shown in the block diagram of Fig. 9-1. At 

least two active elements are assumed included in A, contributing the 

necessary positive gain. 

Upon the application of an input stimulus, e in , the signal returned 
through the amplifier and feedback network /3 is 

e'„ = /34e ln (9_D 

When pA > 1, the returned signal is larger than the applied stimulus, 
and once the switch is closed, it will reinforce the original signal. The 
closed-loop system is unstable, with build-up continuing until the balance 
of returned signal and input is achieved, either through a decrease in the 
amplifier gain (limiting) or by the destruction of the transmission path 
(active element cut off). 
Balance conditions require that when the switch is closed 

< = 4. (9-2) 

247 



248 



SWITCHING 



[Chap. 9 



For possible switching, the system must have more than one stable point 
satisfying Eq. (9-2). 

Figure 9-2, where we have superimposed the unity gain locus (e' = e in ) 
on the loop transfer characteristic of a typical amplifier, i.e., PAe^ versus 
e iB , serves to illustrate the switching action. If the particular point 
selected to break the system's loop, for the evaluation of this curve, is 
normally restricted to positive or negative voltages (e.g., at the plate of a 
tube), then the whole curve will be shifted along the balance line into 
either the first or third quadrant. Inherent circuit limitations will 
always account for the two horizontal regions. 

Satisfaction of the unity-loop-gain criterion is a necessary but insuffi- 
cient condition for stable operation. Consider point Y (where e' a = e m 



e' o °-A0e in <■ 




Fig. 9-2. Loop transfer characteristics for circuit of Fig. 9-1. 



= 0), which, at first glance, appears stable. The slope of the transfer 
characteristic, evaluated at a particular point, is simply the value of the 
corresponding loop gain &A. At the origin, $A > 1, and thus any slight 
perturbation which might momentarily shift the operating point into 
the first quadrant will be amplified and returned as a much larger dis- 
turbance. The operating point travels Up the transfer characteristic 
from Y to point X, where the balance condition is again satisfied. Here 
the incremental gain f3A is less than 1; it is in fact zero. Any small 
disturbance away from X rapidly damps out, and the operating point 
returns to X, one of the two stable points. Similar reasoning shows that 
the other one is point Z. 

We conclude, from the previous discussion, that the second necessary 
condition for stable switching is that fiA > 1 in the active region and 
that fiA < 1 at the stable point. It also follows that there always must 
be a point of unstable equilibrium between each set of stable points. As 
an exercise, the reader may furnish the proof of this statement. 



Sec. 9-2] plate-grid and collector-base multivibrators 249 

If the active elements were transistors instead of vacuum tubes, we 
would modify the previous discussion and define the regenerative action 
in terms of the essential loop current gain. 

The initial start from any unstable point, such as Y, would be 
caused by circuit noise or some other small perturbation. However, 
in order to switch from one absolutely stable point to the other, we 
must disturb the system at least enough to shift the operating point 
into the regenerative region where &A > 1. This region is easily deline- 
ated by marking its boundary points (j}A = 1). We simply find the two 
points on the characteristic with unity slope (K, L). Thus the trigger 
size required for switching from X to Z becomes 

e T > e K — e x 

To shift back from Z, we must introduce a positive disturbance greater 
than ZL. 

When energy-storage elements are included in the internal transmission 
path, the exponential decay will eventually bring the circuit from the 
limiting to the regenerative region. As a consequence, the multivibrator 
switches between the points satisfying the balance conditions without 
resorting to external triggers. But since the system remains at each 
particular state for a well-defined interval, this only changes the stable 
to a quasi-stable point. 

The flat portions of the transfer characteristic arise as a direct result 
of driving one of the active elements into saturation or cutoff. It follows 
that if we modify the load of the active element by the addition of diode 
wave-shaping circuits (Chap. 2), multiple limiting regions become fea- 
sible. Each diode introduces an additional break into the transfer char- 
acteristics and may thus create a new stable, or if energy-storage elements 
are also included, a new quasi-stable, point. 

Transition from state to state takes some small finite time interval, 
primarily the time required to store or dissipate energy in the parasitic 
elements. In addition, when the base amplifier contains transistors, 
their on-off and carrier storage times will further affect the switching 
speed. These delays are not germane to the system's basic behavior, 
and we shall therefore ignore them in the following discussion. 

9-2. Vacuum-tube Bistable Multivibrator. The circuit of a vacuum- 
tube bistable multivibrator (bistable indicating two absolutely stable 
states) appears in Fig. 9-3. Sometimes this circuit is also referred to as 
an Eccles-Jordan circuit, a flip-flop, or a binary. On being triggered, 
each tube switches between two of its three possible states, full on, active, 
or cut off; the particular two depend upon the adjustment of the circuit 
parameters. 



250 



SWITCHING 



[Chap. 9 



It is easy to show that both tubes are normally in different states 
except during the transition interval when they simultaneously become 
active. Assume that both tubes are on, with E ce , a negative voltage, 
chosen to place them in their active regions. Any slight disturbance, 
such as noise, may momentarily raise the plate current of TV The 
resultant drop in plate voltage is coupled by Ri, R 2 , and the speed-up 
capacitor C\ to the grid of T2. The cross-coupling reapplies the amplified 
signal (now a voltage rise) back to the grid of T\, where it reinforces 
the original perturbation. Thus the regenerative action drives T\ toward 
saturation and T 2 toward cutoff. Switching ceases when one or both 
tubes reach their limit. 

Three possible models exist for each tube, one when the tube is cut off 
(Fig. 9-4o), the second when it is in its active region (Fig. 9-46), and the 




Fig. 9-3. Vacuum-tube bistable multivibrator. 
third when it is saturated (Fig. 9-4c). (Figure 9-4a and c are degen- 
erate forms of the general plate-circuit model shown in Fig. 9-46.) In 
drawing the model holding during saturation, we neglected the effect 
of the slight positive grid excursion. When this term becomes significant, 
the model of Fig. 9-46, with the appropriate positive value of e c , will be 
used instead. 

In attempting to analyze a multivibrator, a circuit where many 
combinations of tube states are possible, where and how do we begin? 
The answer is "anywhere"; any logical initial assumption serves as a 
convenient starting point and, if incorrect, will eventually lead to a con- 
tradiction. One of the limited number of other choices will then be 
the correct tube state. Since both tubes cannot remain in the same 
condition, the only permissible combinations of tube states are those 
tabulated below. We assume that the circuit has been triggered so 
that Ti is the on (active) tube and Tt is the off tube. 

Permissible States of Tubes 

Full on Active 

Full on Cut off 

Active Cut off 



Sec. 9-2] plate-grid and collector-babe multivibrators 



251 



Our object thus is to start, say, by assuming Ti fully on, and then 
to carry the steady-state-circuit analysis through to its logical con- 
clusion. Usually (Ri + Ri) » Rl and the loading of the plate by the 
coupling network may be ignored. From Fig. 9-4c, 



And by superposition 

Ed 



Eui = 


R, + r v Eli 


Ri 


■E„+ n R * 



Eu 



(9-3) 



(9-4) 



Rl ~\- 7t2 

where E b ,i is given by Eq. (9-3). Cutoff for this circuit is 

w — ^*» 

Eia, — 

Now if the actual grid voltage E c i is more negative than E a 
Ti is cut off and Fig. 9-4a is the applicable model for calculation of 



(9-5) 
then 






(e) 

Fig. 9-4. Vacuum-tube circuit models, (a) Cutoff region; (6) active region; (c) 
saturation. 

E bi and E cl . But if > E ci > E m , T2 is in its active region and we 
must use the model of Fig. 9-4b. At the stable point both tubes cannot 
be active; if T 2 is part on, then Ti must be full on, and our original 
assumption as to the circuit state was correct. 

To continue, suppose that in this circuit E e2 < E m (T2 cut off); then 
it follows from Fig. 9-4o that 

^2 ra 1 "i 



E C ] 



■ Ebt + 



E e , 



(9-6) 



.Ri -f- Rt Ri + R2 

But to verify the initial assumption, namely, Ti full on, the solution of 
Eq. (9-6) must be E ci > 0. 

Grid-circuit limiting prevents the grid from becoming more than 
slightly positive regardless of the solution of Eq. (9-6). In the model 
used, the grid loading was omitted in order to simplify the calculations. 



252 



SWITCHING 



[Chap. 9 



The alternative solution of Eq. (9-6), E co < E \ < 0, contradicts our 
original assumption, and therefore the problem must be repeated from 
the now known tube states, T\ part on and T 2 cut off. Because Eq. (9-6) 
was written for T 2 off, the substitution of the value of E e \ given by it 
into the model of Fig. 9-46 enables us to resolve the contradiction and 
find the correct values of e b i and e c2 . 

The switching process, the injection of a trigger turning the off tube 
on or the on tube off, will just interchange the states of the two tubes 
of the symmetrical circuit of Fig. 9-3. In an asymmetrical multi- 
vibrator, all voltages must be recalculated as a completely independent 
problem. The widest separation between the two stable states, and 
consequently the easiest to distinguish, occurs when each tube switches 
between full on and cut off. Under these circumstances the total plate 
voltage swing becomes 



Ae 6l 



S a 



Em, — Et,, 



RlEh, 
Rl + T p 



(9-7) 



Proper multivibrator operation is predicated on our establishing the 
correct grid voltages in each of the two circuit states. Since this depends 

Si 




=-E, 



<?c2i0 




Fig. 9-5. Circuit models under limiting conditions, (a) Both tubes saturated; (6) 
both tubes cut off. 

on E ce , Ri, and Ri, we should determine their interrelationships, at least 
at the limits, as a guide in circuit design. One limit occurs when both 
tubes are always saturated, the other when both are cut off. The circuit 
models holding under the limiting conditions, and where (Ri + Ri) > Rl, 
are sketched in Fig. 9-5. 

In order to prevent complete circuit saturation, from Fig. 9-5o we 
see that 

Ri 



E ce < - 



Ri 



Et, 



(9-8) 



where Eb,i is given by Eq. (9-3). The other circuit limit, found from 
Fig. 9-5&, is expressed in Eq. (9-9). 



Ecc> - 



Ri 



E» 






Eu 



(9-9) 



Sec. 9-2] plate-gkid and collector-base multivibrators 253 

The above approximation holds only for high-/* tubes. Generally, the 
center value of Eqs. (9-8) and (9-9) represents a good choice in that it 
allows for the greatest possible variation in the tube and circuit param- 
eters before the multivibrator will cease functioning. 

Rather than switch the multivibrator from one state to the other by 
first injecting a positive trigger to turn the tube on and then a negative 
one to turn it off, we might much more conveniently apply a train of 



+200 




200 v 



100 



• Triggers | | 



(b) 
Fig. 9-6. (a) Bistable multivibrator triggered by a negative pulse train; (6) resultant 
plate waveshapes. 

only positive or negative pulses. In order to maintain grid-to-grid 
isolation, the pulses are coupled into the tubes through diodes (Fig. 9-6o), 
producing the change in state at the plate of T 2 shown in Fig. 9-66. 
The negative pulse drives the on tube into its active region. The 
positive amplified pulse appearing at the plate is coupled into the off 
tube and pulls it into conduction. For positive trigger inputs, the diodes 
are reversed. 



Example 9-1. In this problem we shall investigate the role performed by the 
speed-up capacitor d in the bistable multivibrator of Fig. 9-6. The circuit is excited 
by negative triggers injected into both grids through diodes. The tube used has 
ft = 20, r, = 10 K, r c = 200 ohms, C„ h = 10 nrf, and C m = 10 n4- 



254 switching [Chap. 9 

From Eqs. (9-3) and (9-7) we see that the plate voltage will swing from 200 volts, 
when the tube is cut off, to 100 volts, when the grid is driven slightly positive. To 
prevent false triggering, the grid of the off tube should be maintained somewhat below 
cutoff, say 10 volts below the cutoff value of — 10 volts. Assuming that R x + R t » 
Rl, then with the grid of the off tube at —20 volts, the drop across R 2 must be 80 volts 
(Fig. 9-4c). Since the plate of the on tube is at 100 volts, the drop across R t will be 
120 volts. Hence 

Ri _ 120 

Rt 80 

and we can, quite arbitrarily, choose ijj = 0.5 megohm and Ri = 0.75 megohm. 

We must now check that the on tube is saturated when these coupling resistors are 
used. Substituting the circuit values into the model of Fig. 9-4o, the grid voltage of 
Ti becomes 

e n = 20 volts 

Of course, this calculation neglected the positive-grid limiting and ea would actually 
be only a fraction of a volt. 

When the external pulse is injected, the cutoff grid of T 2 will charge from —20 volts 
toward the new value established by the input trigger, with the time constant 

t« = Ri || RiC ek = 2.85 fxsec 

Once the tube becomes active, the input capacity seen looking into the grid becomes 

C in = C, h + C„(l - A) = 10 + 10[1 - (-10)] = 120 ««f 

Thereafter the charging continues with the much longer time constant of 

t 6 = .Ri || Rid* = 34 Msec 

If the input pulse is extremely narrow, the net change in the grid voltage may be 
insufficient to maintain the switched tube states after the trigger is removed; the 
multivibrator will subsequently return to its original condition. 

By shunting Ri with C lt the coupling network is changed into the compensated 
attenuator discussed in Sec. 1-9. Usually, for the fastest possible switching, it is 
adjusted slightly overcompensated. However, for the purpose of this discussion, 
assume that CiRi = RiC ia or that 

C : = 80 vrf 

This capacity is far from critical, and in an actual circuit it may vary from 25 to 
200 niii. The switching time constants now become, from Eq. (1-34), 

1 '* R, = 0.045 jusec 



0\ + Crt 



R, = 0.245 j«sec 



where R, is the 5-K source impedance seen looking into the plate of the active tube. 
We see that the presence of Ci reduces the critical time constant by a factor of more 
than 100. 

The above argument attempted to show, in a relatively qualitative 
manner, that Ci speeds up the switching process and permits operation 
with quite narrow trigger pulses. Many factors are changing during 



Sec. 9-3] plate-grid and collector-base multivibrators 



255 



the interval considered: one tube is turning off; the other tube first 
becomes active and then its grid rapidly saturates. Therefore no attempt 
will be made to define the required pulse width. We can, however, say 
that it should be somewhat wider than the calculated time constant t£ 
but very much narrower than the pulse needed in the uncompensated 
circuit. 

The bistable multivibrator is a device widely used in digital computers 
and in control equipment for counting. Each unit registers two counts : 
the first input trigger turns off the tube, and the second one turns it 
back on. Thus, if we differentiate the plate waveshape (Fig. 9-66), we 
shall derive only one negative trigger for every two applied. By coupling 
the output pulse to another bistable we count down by four, and finally 
the cascading of n circuits allows counting by 2", yielding one output pulse 
per 2 n inputs. Various feedback arrangements, which preset the cas- 
caded binary chain, will present a single output pulse for any desired 
count. 

9-3. Transistor Bistable Multivibrator. A collector-base-coupled mul- 
tivibrator is the transistor equivalent of the plate-grid-coupled multi- 
vibrator previously discussed. The mode of operation is now determined 




Fig. 9-7. Transistor bistable multivibrator. 

by the base current flow, and therefore, for ensured regenerative switch- 
ing, the loop current gain must be greater than unity. If we keep these 
specific features in mind, then this circuit is also amenable to the argu- 
ments employed in Sec. 9-2. 

Assume that Ti, of the transistor bistable of Fig. 9-7, is in its active 
region. (The circuit shown uses n-p-^n transistors; for p-n-p transistors 
all polarities are reversed.) Any increase in its base current causes a 
proportional increase in collector current. But this is reflected as a 
decrease in the base current of T 2 , where it is amplified and finally 
reapplied as an increase in the base current of 2*1. The original current 
change has thus been multiplied by the total loop current gain. Eventu- 
ally the circuit limits with either T\ full on or T 2 cut off. Since the 



256 



SWITCHING 



[Chap. 9 



speed-up capacitor couples the full change of collector current into the 
other transistor's base, the build-up process is extremely rapid. And 
as in our analysis of the vacuum-tube circuit, transition time will be 
ignored. 

A separate model might be drawn for each of the three regions of 
transistor operation. If, as is normally the case, R c is very much smaller 
than the transistor output impedance and if Ri || Rz is very much larger 
than the small input impedance, then the models become extremely sim- 
ple. Figure 9-8a shows the model holding for cutoff, 9-86 the one defining 
the active region, and 9-8c that holding under saturation conditions. 

For the solution of this circuit, we shall start by assuming Ti cut off. 
The model of Fig. 9-8o gives the base current of Ti as 



IbZ 



Eu 



+ 



E c < 



(9-10) 



Ri + R c Ri 

Models for the active and saturation region must agree at the bound- 
ary. From Fig. 9-8c, we see that the collector-emitter voltage of a 
saturated transistor is zero. Substitution of this limit into the model 
of the active region (Fig. 9-86) results in 



or 



= Eu, - 

Ebb 



(iibRc 



i h > 



PR. 



(9-11) 



Equation (9-11) expresses the bounding condition for maintaining a 
transistor in saturation. 






l&bb 








>Rc 






c 








0h\Q 




> Rl 




bo— 
e 


[ Rz 


Err 


I 




6o 



(a) (b) 

Fig. 9-8. General transistor models, (a) Cutoff region; (b) active region; (c) satura- 
tion region. 

A comparison of Eqs. (9-10) and (9-11) determines the actual state of 
T 2 and enables us to choose the appropriate model. E c i and hi or E iel 
are now calculated, and the initial assumption checked. 

If T\ is not cut off, the contradiction must be resolved by a new solution 
from the other possible starting point, T\ active and r 2 full on. It is, of 



Sec. 9-3] plate-grid and collector-base multivibrators 



257 



course, necessary to modify the general models of Fig. 9-8 so that they 
satisfy the now known transistor states. In particular, we must connect 
the base junction of the coupling network in Fig. 9-86 to ground when 
solving for E cl and hi. 

The limits of E cc or the coupling resistors are found at the two bound- 
aries, complete saturation and complete cutoff. From Eq. (9-10), the 
minimum value of E cc for ibi > is 



E a > - 



Ri 



Ri + R, 



En, 



(9-12) 



The maximum value of E ce , above which both transistors are always 
saturated, will be found directly from Eq. (9-11) through recognizing 
that, under this condition, the total base current must be supplied by 
E ec . Therefore 

Rz 



E c , 



< m Ebh 



(9-13) 



To ensure that one transistor is driven into cutoff while the other 
remains in the active region, much more stringent limitations must be 
placed on E cc [Eq. (9-14)]. If we 
assume that T 2 is on and solve for 
the condition that the voltage at 
the base of T\ must be less than 
we obtain 



zero, 



E„ < 



R2 w 
R[ Ec ' 



(9-14) 




Positive trigger inputs 
Fig. 9-9. Simple transistor multivibrator 
using one power supply and switching 
between full on and cutoff. 



where E ci is the collector voltage 
of the conducting transistor. 

It should be noted that with T 2 
full on {E c2 = 0), Ti is cut off for all 
Ecc < [Eq. (9-14)]. By setting 
E cc = 0, only one power supply is 
required instead of two. Furthermore, since the multivibrator switches 
between full on and full off, current never flows in R 2 and it also may be 
omitted. The very simple multivibrator of Fig. 9-9 results. 

The stable-state current and voltage values for this circuit, with T\ 
cut off and T 2 full on, are 



in — %i = 
e„2 = e c i = 



En 



R c + R 

REtb 
R c + R 



258 switching [Chap. 9 

But in order to ensure the saturation of T*, t&2 must satisfy the condition 
given in Eq. (9-11): 

. _ Etb Ebb , a . -. 

l " - wtr - m (9_15) 

The solution of Eq. (9-15) establishes the maximum value of B as 

R < (P - l)Rc (9-16) 

By treating this circuit as a current amplifier, we shall now demonstrate 
that the condition described above is the very one necessary for guaran- 
teed regenerative switching. Suppose that the loop is broken at the base 
of Tx and that point f is shorted to ground. Then the loop current gain, 
from an input of T\ to point f , is 



£ = A < = (m) v (9 - 17) 



For regenerative operation A c > 1, and thus 

R < (/3 - l)R e 

which is identically the condition previously given by Eq. (9-16). A 
similar inequality must also be satisfied in. the first multivibrator dis- 
cussed in this section. 

The change of circuit state, on to off and back on again, is effected by 
positive triggers injected into both bases through diodes. They drive 
the off transistor into the conduction region, and the resultant of the 
amplified pulse and the original triggers turn the on transistor off. Nega- 
tive triggers will also work, but the diodes would have to be reversed. 

Collector waveshapes, except for voltage values, are of identical form 
with those of the vacuum-tube multivibrator of Fig. 9-6. The maximum 
swing will, of course, be much smaller, and the minimum voltage value 
can now go to zero, but the two circuit states are still clearly delineated, 
one at zero and the other close to I?». 

9-4. A Monostable Transistor Multivibrator. The introduction of a 
single energy-storage element in the regenerative transmission path 
creates a circuit having one stable and one quasi-stable state. If we 
examine the resultant monostable circuit of Fig. 9-10, we note that 
since the base of T 2 is returned to En, this transistor must be on, either 
in its active region or saturated. The polarity of E cc is always opposite 
to that of Eu, and it is relatively easy to ensure that 7\ will be cut off. 

In the particular multivibrator shown, p-n-p transistors are used as the 
active elements. Em, will therefore be negative, and E cc positive. If 



Sec. 9-4] plate-grid and collector-base multivibrators 



259 



n-p-n transistors were used instead, all voltages and currents would be 
reversed. 

The quasi-stable circuit state, Tz cut off, is contingent on the charge 
in the coupling capacitor maintaining e^ > 0. However, a discharge 
path exists and, regardless of the initial condition, C must decay with 
eui subsequently reaching zero. At this point Tt turns back on. Regen- 
eration turns Ti off, and the multivibrator is back in its one "mono" 
stable state. 

Analysis of this circuit will proceed on the assumption that it is 
designed for the maximum possible collector voltage variation, i.e., 
both transistors driven between saturation and cutoff. If any particular 
answer does not support this contention, then the problem must be 
re-solved by drawing new models that will yield consistent results. 




Fig. 9-10. Collector-base-coupled monostable multivibrator using p-n-p transistors. 
Eu is negative and E, c positive for proper biasing. 

The stable circuit conditions are (from the models of Fig. 9-8) 



T 2 on: 



c c i = Ebb 
e C 2 = 



e b ei(0r) = 
42(0-) = 



Ri -\- Rz 

Ebb 

R 



E cl 



If Ti is to be saturated, t» 2 must also satisfy Eq. (9-11), leading to the 
following restriction on R : 

R < 0R C (9-18) 

If this inequality is not satisfied, then, in its normal state, T 2 will be 
active instead of being saturated. 

A negative trigger injected at the base of 7\ turns this transistor on, 
and the change in its collector voltage is immediately coupled through 
C to the base of T 2 , turning it off. The circuit has now entered its quasi- 
stable region, and the new models needed to define operation are those 
given in Fig. 9-11. 



260 switching [Chap. 9 

We determine the time elapsed before the circuit can return to its 
stable state from Fig. 9-1 la. As the right-hand side of C charges from 



31 
6i 



t c i 



c 



b 2 

o—VW-i 
R 



C 2. 



e 6«2 



■Efcfc-^- 



i 



R c 



«CZ 



R 2 



(a) (b) 

Fig. 9-11. Circuit models holding under quasi-stable conditions, (a) Collector of Ti 
to the base of Tt; (ft) collector of Ti to the base of 7*1. 

its original value of — En, toward En,, the equation of the base voltage of 
Ti becomes 

e M (t) = E» - 2JS?»e-" T » n = RC 

At i = h, e b 2 = 0, and T 2 again conducts. Substitution of the boundary 
value into the equation defining en(<) results in 



h = ti In 2 



(9-19) 



To guarantee the saturation of Ti during the unstable period, the follow- 
ing relationship must be satisfied [from Fig. 9-116 and Eq. (9-11)]: 



_ En, , Eec -> En, 
R c + Ri Ri PRc 



(9-20) 



After the circuit reswitches and recovery begins, the models of Fig. 9-11 
are no longer valid. We must consider the new problem posed in Fig. 
9-12, where C recharges toward Eh, from its initial value of zero. The 

collector of Ti recovers toward its 
<?! *°~ b z stable state, also Ew, with the recovery 

I *^f* | l» time constant T2 = R C C 

1 " e cl Within four time constants, re- 

covery is virtually complete. Since 
we wish the recovery time to be small 
compared with the output pulse width, 
R c <K R. This multivibrator is also 
open to simplification by omitting R% and E cc just as discussed in Sec. 9-3. 
Their presence aids in maintaining r 2 well below cutoff while the circuit 
is in its normal state and consequently prevents false triggering by small 
noise pulses. 

Circuit waveshapes are shown in Fig. 9-13, and we might note that 
the output pulse having the best shape appears at the collector of TV 



I 



i 

Fig. 9-12. Recovery circuit. 



Sec. 9-5] plate-grid and collectok-base multivibrators 



261 



Since this point is isolated from the single RC timing circuit, external 
loading, introduced by the coupling to the next stage, will not affect 
the pulse duration. If we differentiate the output pulse, the resultant 

t 



B,+fl 2 Ecc 



t 
eel 



c 




tOE bb t 



t 



t 

E bb 

Fig. 9-13. Monostable multivibrator waveshapes for the p-n-p circuit of Fig. 9-10. 

positive trigger is delayed from the applied input by the pulse duration t\. 
Therefore the monostable multivibrator may be used as either a pulse or 
a delay trigger generator. 

9-5. A Vacuum-tube Monostable Multivibrator. Even a perfunctory 
inspection of the monostable of Fig. 9-14 indicates that in its normal state 



+200 




=r=C, 



Fio. 9-14. Vacuum-tube monostable multivibrator. 

T 2 is fully on (the grid returned through R to Ey, will be slightly above 
zero, and the plate will be slightly below its saturation value of 100 volts). 
Proper choice of E m will bias Ti well below cutoff and yet allow it to 
switch fully on when the circuit is triggered into the unstable state. 



262 switching [Chap. 9 

If we neglect both the small positive grid voltage and the loading of 
Rl by the resistive-coupling network, the grid of the off tube will be at 

Rt in i Ri 



e„i(0-) 



E b ,+ 



E c , 



-20 volts 



(9-21) 



R\ ■+• Ri Ri ■+■ Ri 

which is below the cutoff value of — 10 volts. This tube's plate is, of 
course, at Em,. The only other unknown initial condition is the voltage 
across C, which remains invariant across the transition from the stable 
to the unstable state. Since one end of C is returned to ground through 





(a) (b) 

Fig. 9-15. Quasi-stable state circuit models, (a) Plate of 7\ to the grid of T s ; (6) 
plate of r 2 to the grid of T,. 

the conducting grid of 7 1 2 and the other end is connected to the plate 
of the off tube, 

e,(0~) = e„i(0-) - e c ,(0-) Si E» 

After a positive trigger at the grid of T\ switches the circuit into its 
quasi-stable state, the models defining the behavior become those shown 
in Fig. 9-15. Figure 9-15a is the more important of the two because it 
contains the circuit's single energy-storage element. The model can be 
further simplified by replacing the tube with its Thevenin equivalent — 
a 100-volt source {E hB ) together with a 5,000-ohm resistance (Rl \\ r p ). 
This approximation neglects the slight positive grid excursion that will 
lower the plate to somewhat below Et,. By using the reduced circuit 
we can now solve for the grid voltage of T*. 

2Ebb — Et, 



e* 2 (0+) = En, - 



R 



(9-22) 



R + Rl\\ r p 
But since R~2> Rz,\\r p 

e c2 (0+) Si -(#«. - Et.) = -Si = -100 (9-23) 

where Si, the total plate voltage swing of Ti as it switches from cutoff to 
saturation, is also given by 

Rl 



Si = 



r p + Ri 



Em 



(9-24) 



The grid starts recovering from — 100 toward 200 volts with the time 
constant 

n = (R + Rl II r„)C S RC = 56 Msec 



Sec. 9-5] plate-grid and collector-base multivibrators 



263 



Once it reaches the cutoff value of — 10 volts, T t becomes active and the 
circuit regeneration will rapidly turn T\ off and T% fully on. Conse- 
quently, the duration of the quasi-stable state, as evaluated from the 
exponential-response equation, is 



t\ — t\ In 



Ebb + <Si 



E» + 



Eu 



20 /isec 



(9-25) 



Since, at least to a first approximation, all the voltage terms of Eq. (9-25) 
are linear functions of Ebb, we expect that the pulse duration will be 
relatively independent of any variations in the plate supply voltage. 
Both n and r P will vary with the current flow, n only slightly but r p 
drastically. As a consequence Si will vary somewhat with E bb , and so 
will the pulse duration. 

The final charge on C at the instant before switching back to the stable 
state is 

e t (h) = E b . + — = 110 volts 

This value must be used to calculate the initial value of the recovery 
waveshape from the model of Fig. 9-16. Since the transition occurs at 



■=-E, 



(a) ~ - <b) 

Fig. 9-16. Recovery models for plate-grid-coupled monostable multivibrator, (o) 
pi to gs; (6) ps to pi. 

— Ebb/ ii instead of at zero, during the switching process the tube goes 
directly from cutoff to saturation. As a result, jumps appear at both the 
grid of T 2 and the plate of Ti. From Fig. 9-16o, 





E, 



c2,n 



but since r c <K R L , 
E, 



= Ebb - (E h . + Ebblv) 
Rl + U 



-Tk( Sl ~ir) = 2 - 2 volts (9_26) 

We might note that the voltage term in Eq. (9-26) is simply the total 
change of loop voltage due to the change of the circuit state and that it 
divides proportionately across r e and R L - The total jump at the grid of 
12.2 volts (from -10 to +2.2 volts) will also appear at the plate of T ly 



264 



SWITCHING 



[Chap. 9 

raising its voltage from 100 to 112.2 volts. After the jump, recovery is 
rapid — back to the initial steady-state value with the fast time constant 
of 

r% = (Rl + r c )C^ R L C = 1 Msec 

To obtain the recovery waveshapes at the plate of T 2 and the grid 
of T\ we must treat the model of Fig. 9-166. These are most easily 

„ 200 

t " 





t 

e cl 
Bci(O-) 
















-10 




> 


-20 






200 v 
100 v 


^ 






78 v 


22vj 


S** 1 






t 


12.2 V 




toE bb 






r^i2 »- 


-10 v 


A£ c2 - 


l..---^ * 


-100 v 













■E|>2 mm 



"c2 max 

o 
£» 



-Si 
Fig. 9-17. Waveshapes of plate-grid-coupled monostable multivibrator of Fig. 9-14. 

found by imposing the effect of the positive grid voltage on the quiescent 
values. The 2.2-volt grid excursion is amplified (A = — 10) and reflected 
into the plate of T 2 , causing the voltage to drop 22 volts below the nom- 
inal value of 100 volts: 

hRl 



Eb: 



— Ebs — 



■E, 



78 volts 



(9-27) 



t v + Rl 

The plate now recovers toward E h , at a rate controlled by the grid circuit 
charging. In addition, this large drop is coupled through the speed-up 
capacitor to the grid of 2\ and can only aid in driving it to well below 
cutoff. 



Sec. 9-5] plate-grid and collector-base multivibrators 265 

Equation (9-27) may yield a negative answer for J5 iimln . Obviously, 
this must be rejected since the model representation has led us to an 
impossible solution. Once the tube is driven into the high-grid-current 
region, both r, and r c change markedly from the constant values assumed 
in drawing the piecewise-linear models, and these variations, if taken into 
account, would resolve the contradiction. 

The waveshapes shown in Fig. 9-17 are of the same general form as 
those found for the transistor monostable, and if we had accounted for 
the small base region (about 200 mv), the transistor circuit also would 
have exhibited some very small overshoots. The excursion below E b „ 




Fig. 9-18. Firing-line intersection by charging exponentials, 
appearing at the plate of T 2 , can be easily eliminated by incorporating a 
plate bottoming diode which will conduct and limit the plate voltage to 
some value slightly above Eu- 

As an alternative to the circuit of Fig. 9-14, we might return R to 
ground instead of to Ew The switching sequence still remains the same, 
with the same size plate swing and time constant; the grid, however, now 
charges toward zero. The conduction point —Eh/p is not affected by 
this circuit change, but since the multivibrator switches state closer 
to the final value, the pulse duration will be much longer than that given 
byEq. (9-25): 



a 



T\ 



In 



Etb 



(9-28) 



and for a high-/* tube, which fires almost at zero, this sweep takes approxi- 
mately four time constants. 

Suppose that we adjust the RC time constant of the particular multi- 
vibrator having its grid returned to zero so that its period is identical 
with that of the original circuit (Fig. 9-18). The curve charging toward 
zero (1) crosses the — 2J»/m line at an extremely shallow angle, in fact 
almost horizontally. This is in sharp contrast to curve 2, which expo- 
nentially charges toward Ebb and which intersects the firing line closer 



266 



SWITCHING 



[Chap. 9 



to the perpendicular. Any noise introduced into the system effectively 
lowers the cutoff line to the dotted line of Fig. 9-18. Curve 1 fires 
quite a bit prematurely (at <„), while curve 2 is only slightly affected 
(fires at fe). Thus we conclude that by returning the grid to the highest 
possible voltage, the pulse duration is made much less susceptible to 
random disturbances. 

9-6. Vacuum-tube Astable Multivibrator. The insertion of a second 
capacitor in the transmission loop, i.e., capacitive coupling from each 
plate to the other tube's grid, makes both circuit states quasi-stable. 
Figure 9-19 shows the circuit of the resultant astable multivibrator. 
If both tubes are assumed to be in their active regions, regeneration will 
rapidly drive one into cutoff and the other to saturation. This state is 
unstable, lasting only until the particular capacitor that is keeping the 
tube cut off discharges, at which point the switching action reverses 




c 2 



X 







Fig. 9-19. Plate-grid-coupled astable multivibrator. 

the tube states: the off tube goes on and the on tube goes off. The 
astable multivibrator is self-starting and free-running. 

We might note that the timing network coupling each plate to the 
next grid is exactly the same as the single timing circuit of the mono- 
stable multivibrator (Fig. 9-14). The astable multivibrator may well be 
considered as two monostable circuits operating sequentially, one trig- 
gered off whenever the other one turns on. The previous techniques are 
readily applicable, and by starting anywhere we shall, within no more 
than 2 cycles, reach the final periodic solution. 

Since the exponential timing after the change of circuit state depends 
upon the conditions existing immediately before switching, probably the 
best starting point is where T 2 is full on, just going off, and T x is full off, 
just going on. At this instant the model holding (Fig. 9-20a) is identical 
with the one defining the stable state of the monostable multivibrator 
(Fig. 9-15a). In our initial analysis, so as not to confuse or conceal the 
basic simplicity of the behavior of this circuit beneath the mathematical 
manipulations, we shall ignore the effects of any positive grid voltage. 
In fact, we may even limit e c < by simply setting r c - 0. The initial 



Sec. 9-6] plate-ghid and collector-base multivibrators 267 

circuit voltages, from Eqs. (9-21), (9-22), and the assumed starting point, 
are 

—En 



e c i = 



M 

It 
Ri 



e.i = ■£ E», & 



en = E» 
ebs = Eui 



Tpi 



r P 2 + Rl 



En 



These values determine the initial charge across each coupling capacitor 
and therefore enable us to calculate the voltages at each tube element 
just after switching. The models holding appear in Fig. 9-20 with the 
charge indicated. 




i — wv- 



— E u 



Pi 



C, 



£*.+- 



I 



1 

(b) • r 

Fig. 9-20. Astable circuit models (Ti on, T t off), (a) Sweep model j)i to g,; (b) 
positive grid recovery model p« to g\. 

As the circuit changes state, the plate voltage of the formerly off tube 
Ti drops from E» to E M . This swing Si is coupled by C 2 to the grid of 
T* and will cut off that tube. Since R* » Rl, the effect of the capacitor 
charging current in determining the plate swing is insignificant. Figure 
9-20a shows the timing circuit holding during the unstable interval. 
The grid of T 2 recovers from — Si toward En, with the tube turning 
back on when it reaches —E^/y- The time required is 



ti — Ti In 



Ey, + Si 
En, + Etb/n 



Ti = Rid 



(9-29) 



When the multivibrator reswitches, the sequence of events repeats in 
the network coupling the plate of Ti to the grid of TV 



ti = n In 



Eu + Sj 
Ebb + Ebblp. 



ti Si RiCi 



(9-30) 



The two waveshapes generated are displaced in time from one another, 
one starting when the other ends (Fig. 9-21). Thus the total period is 
the sum of the times given in Eqs. (9-29) and (9-30). If a square-wave 
output is desired, the circuit is made completely symmetrical and h = U. 

Up to this point we have concerned ourselves only with the circuit 
behavior during the off period and we have ignored anything that 
happened at the grid of the on tube. When T t switches on (Fig. 9-20b), 



268 



SWITCHING 



[Chap. 9 



the change in the grid circuit and the introduction of a finite value of 
r c produce a positive grid jump [Eq. (9-26)], with a subsequent recovery 
back to zero. 



e,i(<) ^ tf.we-"* 



RlCi 



(9-31) 



Normally, r 3 <3C t 2 and the recovery of Ti is completed within a very 
short period compared with the off time of T 2 . The grid jump also 




Fig. 9-21. Waveshapes appearing in the plate-grid-coupled astable multivibrator. 



appears amplified at the plate of T t and helps drive the other tube 
off. Thus two driving source components must be considered in calcu- 
lating the exact response of the grid circuit of T 2 , the simple plate drop 
from Eu, to E b ,i (previously considered), and the amplified exponential grid 
decay of Eq. (9-31). 

Since the positive grid recovery of Tx is assumed complete while the 
grid of the off tube is still well below the cutoff value, there will be but 
slight modification needed as to the actual duration of the off period. 



Sec. 9-7] plate-grid and collectoe-base multivibbatoes 



269 



It seems that a reasonable approach is to simply amplify the grid recovery 
waveshape [Eq. (9-31)] and superimpose it on the sweep waveshapes 
which were found by ignoring the positive grid jump. This is indicated 
by the broken lines of Fig. 9-21. If we should require a more exact 
answer as to the sweep duration, we would have to solve for the response 
of the appropriate plate-to-grid RC circuit to the applied driving function, 
— Si + Ae c i(t), where e c i(t) was defined in Eq. (9-31). 

9-7; Transistor Astable Multivibrator. The circuit of a symmetrical 
astable multivibrator using n-p-n transistors appears in Fig. 9-22. Each 
transistor sequentially switches between full on and full off and behaves 
in a manner similar to the single unstable state of the monostable multi- 
vibrator. The circuit waveshapes are also basically those of the mono- 
stable (Fig. 9-13), with the alternate transistor states displaced by half 
of the total period (Fig. 9-23). 




(3-50 



Fig. 9-22. Transistor astable multivibrator. 

Once T2 goes full on, the base of T\ is driven to —10 volts and the 
collector of Ti recovers to Em, with the time constant R C C. To ensure 
complete recovery before the circuit reswitches, the half period, iden- 
tically that given in Eq. (9-19), must be longer than the recovery time. 



RC In 2 > ±R C C 



(9-32) 



If, in addition, we should want to ensure that the on transistor remains 
saturated over the complete half cycle, then a further limitation, defined 
by Eq. (9-18), must also be imposed on R. Combining Eqs. (9-18) and 
(9-32), we see that R should be restricted to the range 



PR,> R> 5.8R C 



(9-33) 



In the circuit of Fig. 9-22, R must lie between 5.8 and 50 K. Cur- 
rently available transistors have /3 values of 50 or more; hence the 
limitation on R is not particularly severe. Since C is the same in both 
charging paths, the larger the size of R, consistent with Eq. (9-33), 
the smaller the percentage of the half period devoted to the collector 
recovery exponential. The waveshapes of one transistor are shown in 



270 



SWITCHING 



[Chap. 9 



Fig. 9-23; those at the other transistor are identical but displaced by a 
half period. 



1 u 

'61 

En, 
i Ebb 

cl 






















( 


_J 


£ 


1 1 
1 1 
1 1 
1 1 




=.fl c C 


1 /r- 

1/ 

1/ 


*- 



Fig. 9-23. Waveshapes of transistor astable multivibrator (taken at transistor T t ). 

Example 9-2. By setting R - 75 K in the circuit of Fig. 9-22, Eq. (9-33) is not 
satisfied and the transistors switch between their cutoff and active regions. This 
mode of operation is somewhat different from that previously considered for the 
transistor, but since the basic behavior remains the same, we may use the standard 
method of analysis. 

As our arbitrary starting point, we shall assume that Ti is off, on the verge of switch- 
ing on, and that T 2 is in the opposite state. In the initial discussion the contribution 
of the charging current through Ct and the base of T t will be neglected, and we shall 
further assume that the total bias current is Eu/R. When Ti turns on, its collector 
falls from 10 volts to 

•Eci(0+) "Em,- piiRc =» 10 - 0;4t? 1K= 2.5 volts 

to Iv 

The base of T 2 falls by the same amount and immediately starts charging from 
—7.5 volts toward 10 volts with the time constant 

ti = RC = 75 msec 

Since the turn-on point is zero volts, all the information necessary to calculate the 
switching time is now known. The half period is 

h - 75 In 10 + 7 5 S 42 msec 

In the previous calculations we neglected the component of the base current of T> 
contributed by the recharge of Cj. This current flows through the 1-K collector load 
resistor, starting from a peak of 

, 7.5 volts „ , 
im = 5 = 7.5 ma 

tic 

and decays toward zero with the fast time constant 

T 2 = RcC = 1 msec 



The extra current flow drives the transistor into saturation for a portion of the 
cycle. This phenomenon is almost identical with the positive grid jump in the vac- 



Sec. 9-7] plate-grid and collector-base multivibrators 



271 



uum-tube circuit where the fast overshoot brought the plate voltage below Et,. Just 
as in that case, Tt<H n and the extra recovery exponential will not influence the sweep 
timing. It may simply be superimposed on the normal sweep as shown by the heavy 
broken lines in Fig. 9-24. 



10 



2.5 

















t 


f 


A 




i 


• 




42 msec 

! ! 

i i 
* i 




t 


: 


a 


s 


. 


A 






1 

r 





-7.5 
-10 

Fig. 9-24. Waveshapes for the multivibrator of Example 9-2. 

Under the conditions of this problem, the initial drop, and hence the half period, are 
dependent on the value of 0. A 10 per cent reduction, from 50 to 45, reduces the size 
of the voltage drop to 6.75 volts and the sweep time to 39 msec. If the transistor 
switched between cutoff and saturation, as it does when generating the waveshapes of 
Fig. 9-23, then the period is determined solely by the BC network and not by the 
parameters of the active circuit element. 

Incomplete Circuit Recovery. We now propose to consider the addi- 
tional complexity introduced in calculating the duration of the half 
period, if the collector recovery is not complete when the multivibrator 
switches states. The symmetrical astable is relatively easy to treat, 
once it builds up to periodicity, since the waveshapes of both transistors 
are identical. The base of one transistor is driven off by the drop in the 
collector voltage of the other transistor as it switches full on. At the 
end of the unknown half period h, the collector voltage has risen only 
from zero to the value found from the recovery exponential [Eq. (9-34)]. 
(Also see the waveshape in Fig. 9-25.) 



«.(*i) = -Ml - e-''"') t 2 = R C C 



(9-34) 



Therefore the off transistor is driven only to — e c (h) instead of to — E». 
The equation denning the rise of base voltage becomes 



e h {t) = E» - [E* + e.(«i)]« rM * 1 n = RC 



(9-35) 



A change of state again occurs when e&(<i) = 0. But the elapsed 
time depends on the initial drop [Eq. (9-34)], which itself is determined 



272 



SWITCHING 



[Chap. 9 



by the unknown half period. The interrelationships of the two equations 
are illustrated in Fig. 9-25, especially by comparing it with Figs. 9-23 
and 9-24. 

The above argument has furnished us with two transcendental equa- 
tions [(9-34) and (9-35)] which must be simultaneously solved for the 
unknown half period h. One standard method of solution is to graph 
e e i as a function of various values of ti, as individually found from each 
equation. The intersection represents the unique solution for both e cl 
and <i. 



„ 












e c (h) 




^*~ 




,.-> 








/ 




t 
«cl 






Az 


-AE cl 


f 






i ! 




't 


o 


« t x , 


i 

■p <! > 






t 


t 

e M 




AE cl 






/ 










/ 


-e c (h) 
~Ebb 




^ 






f 





Fig. 9-25. Collector and base waveshapes due to incomplete recovery. 

In the special case of a symmetrical transistor multivibrator, by sub- 
stituting Eq. (9-34) into (9-35), we find that the half period h corresponds 
to the solution of 

1 _ 2e-«' /T ' + e -'' /r 'e-'' /T » = (9-36) 

By making the following additional substitutions, 



XT2 



x = e 



,-hhi 



Eq. (9-36) reduces to 



2x + 1 = 



x < 1 



(9-37) 



The restriction on x must be imposed so that the solution of Eq. (9-37) 
for the half period will correspond to real values of time. 

In general, X is not an integer and Eq. (9-37) would have to be solved 
by numerical or graphical methods for the single root lying within 

< xi < 1 



Sec. 9-7] plate-grid and collector-base mtjltivd3Rators 273 

Finally, since xi = exp [ — (ii/n)], the half period is 

ii = n In — 

X\ 

For X large, the recovery time constant ti is very much smaller than 
that of the rise and the root of Eq. (9-37) is located at 

This simply yields the half period for the complete recovery waveshape, 
i.e., the maximum possible sweep duration of 

h = n In 2 

which was given in Eq. (9-19). The second limiting location of the tim- 
ing root appears where r\ = t 2 (i.e., when X = 1). Under this condition, 
Eq. (9-37) may be expressed as 

(x - l) s = 



and the half period is reduced to zero. 



1.0 


, 
















































































0.8 


















































































k 0.6 












































































0.4 


















































































0.2 






























































































































0.2 



0.4 0.6 

1 r 2 



0.8 



1.0 



Fig. 9-26. Normalized half period as a function of the ratio of time constants [Eq. 
(9-37)]. 

A plot of the solution of Eq. (9-37) appears in Fig. 9-26. We can see 
that the recovery exponential has almost no effect on the sweep period 
when r 2 < 0.2ti, but as t 2 approaches the sweep time constant n, the 
sweep time rapidly decreases toward zero. Operation in this mode is 
usually avoided since the slightest variation in circuit parameters has 
pronounced repercussions on the waveshape produced. There are multi- 
vibrators, however, which depend on the incomplete recovery behavior to 
generate extremely fast pulses. In this case, the transistor would 
never be allowed to saturate (Sec. 9-9) and the analysis would be slightly 
more complicated than that given above. 

If the multivibrator is asymmetrical, we shall be faced with four 
transcendental equations, two for each half cycle. The evaluation of the 



274 switching [Chap. 9 

two unknown time intervals and the two starting voltages would clearly 
be most tedious. 

Returning briefly to the vacuum-tube astable, incomplete recovery 
also means that the double exponential grid waveshape, shown by the 
broken lines of Fig. 9-21, persists until the off tube turns on. The plate 
voltage of the on tube reflects the positive grid-recovery exponential 
into the circuit which maintains the next grid cut off. Even in a sym- 
metrical circuit, each of the two equations would involve double-energy- 
storage conditions and would be extremely difficult to solve. 

9-8. Inductively Timed Multivibrators. In the capacitively timed 
circuits discussed above, the regenerative loop was interrupted and the 
quasi-stable state established by cutting off the tube (transistor). This 




(a) (b) (c) 

Fig. 9-27. (o) Inductively timed monostable multivibrator; (6) model holding during 
the stable state ; (c) model defining the timing interval. 

action was voltage-controlled, with the amount of stored energy and the 
discharge path determining the duration of the pulse generated. As an 
alternative, the loop gain can be reduced to below unity by keeping the 
transistor saturated for a controlled time interval. Because this is a 
current-dependent action, we shall use an inductance as the timing 
element. 

Let us consider the monostable multivibrator of Fig. 9-27a, where, 
in the normal state, T\ is saturated. Since its collector voltage is zero, 
T 2 must be cut off. The initial circuit conditions, from the model of 
Fig. 9-276, are 

Ebb 



/n(O-) = 

E cl (0r) = 
E c2 (0-) 



R 3 + R 
/ci(O-) = 



Ebb 

Ri 

E bl (0-) = E M (Qr) = 
Ri 



Rs + Ri 



Ebi 



(9-38a) 

(9-386) 
(9-38c) 
(9-38d) 



Sec. 9-8] plate-grid and collector-base multivibrators 275 

and to ensure the saturation of Ti, fiRi > (R* + Ri). Because Ri S> R%, 
the drop across R t may be neglected in any computations. 

The injection of a negative pulse at the base of T x turns off this tran- 
sistor. The current previously flowing through L must remain constant 
across the switching interval. Il(0~) will now flow into the base of T it 
driving it far into saturation. The current in L immediately starts 
flowing through the path shown in Fig. 9-27c, decaying toward 

Ehb (9-39) 



ill ~\~ Ri 



with the time constant n = L/(Ri + R^). For regenerative switching, 
the circuit must enter the active region by itself. This requires that 
/„, given in Eq. (9-39) , must be less than the saturation value of the base 
current of T 2 ; i.e., it must be below 

/., = H (9-40) 

By writing the required inequality, the condition which must be satis- 
fied may be expressed as /3fl 3 < Ri + Ri- 

Once T% enters the active region, the increase in its collector voltage 
forward-biases jTi, bringing it from cutoff into conduction. Regenera- 
tion completes the switching, carrying Ti into saturation and turning 
T 2 back off. 

The duration of the pulse generated may be found from the exponential 
charging equation. Substituting the initial current [Eq. (9-386)], the 
final current [Eq. (9-40)], and the steady-state current [Eq. (9-39)], this 
time becomes 

h = rx In Z " ~ /c V ( °" ) 0-41) 

The various current limits are indicated in the waveshapes of Fig. 9-28. 
Care must be taken to limit the maximum voltage appearing at the 
collector of T\. On starting, when the coil current switches from the 
collector of T"i to the base of T 2 , this voltage jumps from zero [Eq. (9-38c)] 
to 

e«i(0+) = J.i(0-)fl, = |- 2 E* 

In general, Ri > R t and the peak collector voltage will be several times 
as large as E». 

At the end of the pulse, the inductance must recharge to its initial 
state (the saturation current of Ti). Since Ti is again saturated (Fig. 
9-276), it does so with the relatively long time constant t% = L/Ri, 



276 



SWITCHING 



[Chap. 9 



and because the recovery time is much longer than the pulse width, this 
circuit is only applicable where the wide pulse spacing can be tolerated. 
Thermal instability in a transistor multivibrator is due to the tempera- 
ture-dependent I c o, which also flows into C, increasing the total charging 
current. This reduces the sweep duration, and when attempting to 
generate long-duration pulses, the large resistance used in the capacitive 
timed circuit compounds this instability. On the other hand, if an 
inductance is used as the timing element, the series resistance must be 



■Ri 



Ehh 







\ E bb 

e C 2 





Em 
l f 

iss 








X 


^■^ 










t 








\ t 




\5 


^«— - " — 










c 




't 



Fio. 9-28. Inductively timed monostable multivibrator waveshapes. 

minimized if we wish to have a long time constant. The effects of I c o 
become almost negligible and pulse time stability is ensured. 

Inductive Astdble Operation (?). It is quite difficult to design a simple, 
easily controlled, astable version of the inductively timed multivibrator. 
In the symmetrical circuit of Fig. 9-29, depending solely on the LR 
charging (the coil capacity assumed completely negligible), the time 
constant which maintains the saturation of one transistor is 



Tl = 



Ri + R% 

But the coil recovers with the much longer time constant 

L 



Tj = 



Ri 




Sec. 9-9] plate-grid and collector-base multivibrators 277 

The argument employed with respect to the incomplete recovery in the 
capacitively timed circuit, which led to Eq. (9-37) , also holdsf or this circuit. 
We draw the conclusion that no root corresponding to real time can be 
found for the condition of n < t 2 , and consequently the period of this 
multivibrator must be zero. 

In the special case where the coupling resistor Rz is set equal to zero 
and where Ri is very small, the transistor parameters may play a major 
role in determining the two time con- 
stants of the circuit. During the satu- 
ration interval, the conduction path is 
through the base-to-emitter resistance 
of the transistor. Recovery is through 
the much larger saturation resistance 
of the collector emitter path, and there- 
fore with a faster time constant. Thus 
r2 < t\, and the period may be found 
by solving an equation similar to Eq. 
(9-37). It should again be noted that Fig. 9-29. Symmetrical circuit with 
this multivibrator depends for its tim- inductive energy-storage elements 
ing on the second-order effects of the (not an astable circuit) ' 
circuit and its operation may not be very dependable. 

Several free-running circuits exist that apparently use inductive tim- 
ing for their operation. In all cases, either a second energy-storage 
element is present, making the coil a resonant circuit, or the two coils are 
coupled and the core driven into saturation on alternative half cycles. 
These circuits depend for their timing on other than a single mode of 
energy storage. Their solution is somewhat more complicated than the 
simple multivibrators discussed and is therefore beyond the scope of this 
section of the text. 

9-9. Multivibrator Transition Time. A number of approximations 
were made in the course of our analysis of the multivibrator in the 
interest of keeping the important characteristics of the circuit in the fore- 
front. The major assumption was that the multivibrator switched 
states in zero time, which allowed us to treat only the circuit behavior on 
both sides of the boundary. Calculation of the exact switching interval 
is much too complex for a simple piecewise-linear approach, especially 
since many of the significant terms were thrown out in linearizing the cir- 
cuit. A qualitative discussion will still point out the problems involved 
in the switching process and will lead to the necessary conditions for 
optimization. 

The transition time of a vacuum-tube multivibrator is primarily lim- 
ited by the parasitic capacity and inductance present in the circuit. For 
ideal operation, the change of tube state, i.e., from on to off, should be 



278 switching [Chap. 9 

accompanied by large instantaneous changes in both tube current and 
voltage, but lead inductance limits the time rate of change of current and 
stray capacity will prevent any instantaneous change of circuit voltage. 
The response to a change of state is quite similar to the transient response 
of the tubes and coupling networks to an applied unit step of grid voltage. 
For switching rates up to about 100 kc this is not very important; but 
if we try to design multivibrators to operate at several megacycles, the 
transition time is often the limiting factor. 

The conclusions to be drawn from the above discussion are that the 
switching time may be improved by using small plate load resistors, by 
keeping the stray capacity low, and, if necessary, by high-frequency 
compensation. Pentodes, with their improved high-frequency response, 
are also occasionally used in high-speed multivibrators. 

Transistor multivibrators normally operate with very small values of 
collector resistance, which makes the effects of the parasitic elements 
relatively unimportant. The basic limitations on switching time are 
due to the properties of the transistor itself. Three factors must be 
considered : 

1. The transit time in the transistor 

2. The cutoff frequency of the transistor 

3. The storage time of the minority carriers 

Current carrier velocity in a transistor, or any other semiconductor, 
is quite slow compared with the speed of electron travel in the high- 
vacuum interelectrode space of a tube. Any abrupt change of the 
external forcing function requires some finite time before it makes itself 
felt as the collector. First carriers must be injected, and then they 
have to travel across the junction. In turning off the transistors, cur- 
rent will continue flowing until the carriers, previously injected, are 
swept out. The transistor on and off times are proportional to the spac- 
ing of the base-collector, base-emitter junctions. With the new diffusion 
techniques of producing thin base films and extremely small junctions, 
the transit time can be made quite short. Special switching tran- 
sistors having on-off times of less than 25 imisec are currently available, 
and improved manufacturing techniques give promise of even further 
reduction. 

The transistor current-amplification factor falls off with increasing 
frequency as 



<*(/) = 



i +;'(///.) 



where f a is the a cutoff frequency. When we use the transistor in a 
grounded-emitter circuit, the transformation £ = a/(l — «) yields the 



Sec. 9-9] plate-grid and collector-base multivibrators 279 

expression for the /3 variation with frequency 



0(/) = 



l+j[///.U-«)] 



Thus the /3 cutoff frequency is /„(1 — a), and since a is only slightly less 
than 1, //j is a very small percentage of /«. In poor transistors, the /? 
frequency may even be as low as 5 kc, but in the better ones it rises to 
100 or 200 kc. At some sufficiently high frequency the reduction in 
will cause the loop gain to drop below unity. There can no longer be any 
regenerative action above this point. If excessively narrow pulses are 
used for switching, the loop amplification of their high-frequency com- 
ponents may be insufficient to ensure a change of circuit state. For 
high-speed switching, we must always look for transistors having a high 
a cutoff frequency. 

Operation in the transistor saturation region is accompanied by the 
injection of minority carriers into the base region from both the collector 
and the emitter. When we try to turn off the transistor, forward current 
continues flowing until these carriers are swept or diffused out. But this 
takes an appreciable time, usually about ten times as long as the on-off 
time of the transistor in its active region. Therefore the transistor must 
never be allowed to saturate in high-speed switching circuits. 

If we attempt to prevent transistor saturation by increasing the size of 
the bias resistor, so that R > f}R c [Eq. (9-18)], then both the collector 
drop and the unstable period become dependent on 0. In addition, 
during the collector recovery, the charging current also flows through the 
base emitter circuit of the on transistor. Unless the collector voltage 
change is severely restricted to allow for the additional base current flow, 
the transistor may be driven into saturation for a portion of the cycle. 

Our object, then, is to prevent the transistor from bottoming. One 
method which might be used is to connect a diode from the collector to a 
small external bias voltage. Once the collector falls to this voltage, the 
diode conducts and maintains the transistor in its active region. Only 
diodes which have fast recovery time may be used; otherwise the storage 
problem will simply be transferred from the transistor to the diode. 

A second possible circuit configuration involves shunting the collector 
load by a Zener diode (Fig. 9-30o). This diode fires when the drop 
across R c equals E, and subsequently operates in a region where there 
are no carrier storage delays. The load resistance is now shunted by 
r„ reducing the loop gain well below the value needed to sustain regenera- 
tion. Figure 9-30b illustrates the abrupt change in the path of operation 
once the diode fires. In addition, if the circuit is designed to saturate 
in the absence of the Zener diode, then the change in the collector voltage 



280 



SWITCHING 



[Chap. 9 



is always E z and the sweep period is again independent of the transistor 
parameters. 

Many alternative methods of preventing transistor saturation are in 
common use. They differ only in the circuit location of the diodes and 
in their particular firing points. 




Fig. 9-30. Circuit to prevent bottoming through the use of a Zener diode, (a) 
Transistor collector circuit; (b) transistor characteristic and path of operation. 

9-10. Multivibrator Triggering and Synchronization. The minimum 
trigger amplitude depends on its point of insertion. Once this has been 
decided, we can draw a model of the multivibrator and include the pulse 
generator along with its source impedance. Since we know the loading 
introduced and the change-over point of the circuit, the required pulse 
amplitude is easily calculated. 

As an example, consider the problem involved when we apply a nega- 
tive pulse to the grid of the on tube. It must be large enough so that the 
amplified plate pulse, which is coupled to the grid of the off tube, can 
turn the second tube on. We gain one stage of amplification, but the 
trigger source is loaded by the low impedance of a conducting grid. On 
the other hand, a positive pulse applied to the off grid will have abso- 
lutely no effect until it is large enough to bring the tube into the active 
region. It will always see a very high impedance, that of a nonconduct- 
ing grid. 

Exactly the same problems must be faced in triggering the transistor 
multivibrator except that now the trigger source should be able to supply 
the base current requirements when turning it on or off, as the case may 
be. 

Diodes are almost always used in injecting the trigger pulses, for two 
reasons: first, to decouple the two bases or grids until the application of a 
trigger and, second, to prevent a pulse of the wrong polarity from falsely 
triggering the multivibrator. This might happen if a rectangular pulse 



Sec. 9-10] plate-gbid and collector-base multivibrators 



281 



were coupled to both grids through a small capacitor. It would be 
differentiated, and the positive input trigger, produced from the leading 
edge, would turn the off tube on. The trailing edge of the applied input 
also supplies a trigger, a negative pulse which may now turn the on tube 
off and thus return the circuit to its original state. Decoupling through 
diodes would prevent the second pulse from ever appearing at a grid 
and falsely retriggering the tube. 

We may also inject a synchronizing signal into one or both bases (grids) 
of the astable multivibrator and lock its free-running period to some 
submultiple of the sync frequency. The charging curve would no longer 
be a simple exponential but would have the external signal superimposed. 
The tube conducts when the total grid voltage reaches cutoff (Fig. 9-3 la). 
And as was done in Chap. 5, the sync signal might be considered as 




Fig. 9-31. Sine-wave synchronization of a vacuum-tube multivibrator. 

effectively changing the tube cutoff voltage (Fig. 9-316). If we approxi- 
mate the exponential sweep by a straight line, then all the results of 
Chap. 5 dealing with synchronization may be applied to the astable 
multivibrator. In Fig. 9-31, the broken lines indicate the normal free- 
running waveshapes, while the solid curves are the ones due to synchro- 
nization by the sinusoidal signal e„ 



PROBLEMS 

9-1. Assuming that the amplifier of Fig. 9-1 is a perfect amplifier having a constant 
gain A = 10, design a two-diode feedback network p, such that the system's stable 
points are at e in = ±10 volts and the switching points are at ei„ = ±5 volts. You 
may use any required resistances and voltage sources in addition to two diodes in 
designing the passive network. Let the smallest resistor equal 10 K. Sketch the 
transfer characteristics and the operating line. 

9-2. (a) Draw the loop transfer characteristics for the active network of Fig. 9-32. 
Assume that all amplifiers and feedback networks are ideal over the complete range. 
What are the coordinates of the stable points, and how large must the input pulses 
be for triggering? 

(6) Repeat part o when 0i is changed from 0.1 to 0.01. 



282 



-200 



SWITCHING 

M 



[Chap. 9 



A, — 10 



/. 

<>eo 



/3 2 -0.1 



(3,-0.1 



50v-=- — 20v 

"I T 



Fig. 9-32 



9-3. The bistable multivibrator of Fig. 9-3 employs a 12AU7 with equal plate 
resistors of 20 K. The plate supply is 200 volts, and the coupling network consists 
of Ri = 0.5 megohm and R* = 1 megohm. Evaluate the states of the two tubes 
as a function of E„. Tabulate the results and state which conditions will give 
the greatest output swing. 

9-4. An asymmetrical bistable multivibrator uses tubes having /» = 25, r p =■ 10 K, 
and r« = 1 K. One plate resistor is 40 K, and the other is 10 K. The coupling net- 
works are identical with Ri - R t - 200 K. Ew, = 250 volts, and E„ - -200 volts. 

(a) Sketch and label the waveshape at each plate when alternate positive and 
negative pulses are injected into the grid of T\. 

(b) What are the minimum amplitude pulses required to cause switching? To 
which grid must they be applied? 

9-5. Design a bistable multivibrator using a 12AX7 and two 200-volt power sup- 
plies. The plate swing must be 125 volts as the tube switches from cutoff to full on. 
Specify all resistors and show how this circuit would be triggered by a chain of negative 
pulses. 

9-6. By coupling the cathodes of the two tubes of a bistable multivibrator together, 
as shown in Fig. 9-33, the need for a second power supply is eliminated. For this 
circuit calculate the values of R K and Rl that are required to keep one tube just cut 
off when the other tube just reaches saturation. What is the function of Ck, and 
must it be very large or can it be relatively small? 



+300 v 




Fio. 9-33 

9-7. During the transition from on to off, the behavior of the bistable multivibrator 
may be approximately represented by the model of Fig. 9-34o. In this model C 
includes the stray, interelectrode, and speed-up capacity. By taking the TheVenin 



PLATE-GRID AND COLLECTOR-BASE MULTIVIBRATORS 



283 



equivalent across each capacitor, the model may be simplified to the two-loop net- 
work shown in Pig. 9-346, where e n is the equivalent pulse source inserted for triggering. 

(o) Find the poles of the equivalent network by writing the mesh equations as a 
function of p, setting the network determinant equal to zero, and solving for p t and p s . 

(6) Show that the form of the transient response of the networks is 

e. = -Ae"' + Be*' 

(c) If ft = 100, C =■ 50 put, Rt — 10 K, Bi - 500 K, and the impulse of voltage 
applied is infinitesimally narrow but with an area of 10 - ' volt-sec, find the time 
required for e„ to increase by 100 volts. 






zc 


R'z 






R'l 




1 

I 1 " 


cz 




fcW 


t 


1 




] + 




~| 


T 


1 





(b) 
Fig. 9-34 

8-8. Sketch the waveshapes at the base of r, and the collector of T t from I = 
for the circuit of Fig. 9-35. Initially, T t is cut off. 

9-9. (o) Draw the transfer characteristics and the unity gain line for the multi- 
vibrator of Fig. 9-35. Evaluate the minimum-size trigger (current) necessary for 
proper switching. 

(6) Calculate the maximum-size resistive-coupling network necessary to ensure 
reliable operation. Repeat part a for this condition. 



+20v 



*l 


, 


c 




in 


- _ ,_, 




-10 










t, msec 


1 


2 


3 


4 


h ' 




0-70 



Fig. 9-35 



284 



SWITCHING 



[Chap. 9 



9-10. A transistor having & = 25 is to be used in the circuit of Fig. 9-9. If R„ = IK 
and Eu = 2 volts, specify a value for B that will allow for a ± 50 per cent variation in 
fl between transistors and will still permit reliable operation. Calculate the change 
in collector voltage from state to state. 

9-11. A simple direct-coupled bistable multivibrator makes use of the small back 
voltage which must be overcome before the transistor turns on. In addition to the 
characteristic shown in Fig. 9-36, the collector-to-emitter saturation resistance of the 
transistor is 50 ohms and in the active region = 40. 

(a) Show that the circuit of Fig. 9-36 is a perfectly stable circuit having two distinct 
states. 

(6) Derive the magnitude of the minimum switching pulse. 

(c) Calculate the voltage change at the collector. 

(d) To what value must the collector resistor be changed if we want to ensure that 
the transistor is never driven into saturation? Would the circuit function properly 
under this condition? 



? + 1.0v 




9-12. (a) Show at least three ways of ensuring the triggering of a bistable vacuum- 
tube multivibrator on each pulse. A train of positive pulses or a train of negative 
pulses is all that is available. 

(6) Repeat part a for the transistor multivibrators of Sec. 9-3. 

(c) Discuss the behavior of a bistable if a narrow pulse is coupled into both active 
elements simultaneously through a small capacitor. Consider both polarity pulses 
injected into the control terminals (i.e., the base or the grid). 

9-13. We wish to complete the design of the monostable circuit shown in Fig. 9-37. 

(a) Calculate the value of R c needed to drive T t just to saturation in the stable state. 

(6) What value of C is required to keep the circuit in its quasi-stable state for 
1 msec? 




/3 = 40 



Fig. 9-37 



PLATE-GRID AND COLLECTOR-BASE MULTIVIBRATORS 285 

(c) Sketch the waveforms of the base and collector voltages of both transistors 
when a pulse is applied to the base of Ti at t = 0. Label all break points and time 
constants numerically. 

9-14. The transistor monostable of Fig. 9-38 is triggered at t = 0. Calculate the 
waveshapes at the base and at the collector of T it giving all times and time constants. 

9-16. In Fig. 9-38, the timing resistor R is returned to a variable supply rather than 
to 10 volts. Plot the pulse duration as a function of this control voltage E for the 
range of 1 to 25 volts. 



+ 10v 




Fig. 9-38 

9-16. The monostable multivibrator of Fig. 9-38 is modified by connecting a Zener 
diode which will fire at 6 volts from each collector to Em- This prevents the transistors 
from being driven into saturation and improves the recovery time (Sec. 9-9). 

(a) Draw the resultant circuit. 

(6) Repeat Prob. 9-14 for this new circuit. 

(c) Plot the operating locus of the transistor on the collector characteristics. 

9-17. The monostable multivibrator of Fig. 9-14 employs a 12AX7 adjusted so 
that the grid of the off tube is normally at —30 volts. Calculate the required value of 
Eec if the other components are Rl = 100 K, R — 1 megohm, Ri = 1 megohm, 
Rz = 2 megohms, and E» = 250 volts. Find the required value of C to make the 
output pulse duration 100 /usee. Sketch and label the waveshapes appearing at the 
grid and plate of the normally off tube after a trigger is injected. 

9-18. Design a monostable multivibrator that will generate a 150-volt 2-msec 
pulse at the plate of the normally on tube. Use a 12AU7 returned to 250 volts, 
E„ — — 150 volts, and a +20-volt trigger pulse. Specify all component values and 
show where and how the trigger is applied. 

9-19. Consider the monostable circuit of Fig. 9-39, where the voltage at both plates 
is limited by bottoming diodes. 

(a) Sketch to scale the waveshapes appearing at the plate and grid of Tt after a 
large positive trigger is momentarily applied to the grid of Ti. 

(b) Repeat part o when the timing resistor jBi is returned to ground and adjusted 
to produce the same duration pulse as found in part o. Specify the new value of Ri. 

(c) The tube parameters may be expected to vary by +30 per cent from the nomi- 
nal values. In light of these expected variations from tube to tube, what function 
do the diodes perform? 

9-20. The monostable circuit of Fig. 9-39 is triggered by a pulse applied through a 
diode to one grid. If the internal impedance of the trigger generator is 500 ohms, 
calculate the minimum-amplitude trigger (open-circuit) under the following conditions : 

(a) A positive trigger is injected at the grid of T\. 

(b) A negative pulse is applied to the grid of T t . 

Be careful to check the state of each tube immediately after switching. 



286 



SWITCHING 

?+250v 



[Chap. 9 




7.5 K 



0-21. When the astable multivibrator of Fig. 9-19 is made symmetrical, it will 
generate a square wave. Calculate and sketch the waveshapes at one tube if 2J» =■ 
300 volts, R L = 50 K, n = 70, r p = 20 K, r e = 500, R = 500 K, and C - 0.001 „{. 

9-22. Design an astable multivibrator using a 12AXJ7 and having a plate swing of 
150 volts. Ti should be on for 100 /usee, and T t for 900 /usee. Use a plate supply of 
250 volts. Sketch the plate and grid waveshapes, checking that each tube recovers 
completely before reswitching. 

9-23. We desire to build an astable multivibrator for use as a square-wave generator. 
The output at the plates should be 200 volts peak to peak and should have a d-c level 
of 200 volts and a period of 10 msec. At each plate we insert a limiting diode to 
remove any overshoot which would otherwise appear. If the tube available has 
M = 70 and r„ = 50 K, specify all other circuit components. 

9-24, Consider the application of the complete plate waveshape of Fig. 9-21 (broken 
line) to the grid-circuit timing network of the astable multivibrator. This may be 
represented as shown in Fig. 9-40. Assume that the capacitor is initially uncharged 
and that the output voltage e c is zero at t = 0. The excitation ei is 

e t = —150 - 50e-"^ t 4 - 60 ^sec 

The interval of interest to us is where e c < —10 volts. 

(a) Calculate the exact output waveshape during the interval that the second tube 
is maintained off. What is the length of this interval? Compare this result with the 
approximate solution. 

(6) The recovery time constant t< is increased until the time at which the next 
tube turns on is changed by 5 per cent. What is the ratio of the excitation time con- 
stant t« to the circuit time constant under this condition? Is the approximation 
made in the text valid? 




0.001 »f 2M 

9 |t T — VW- 1 



T 

♦ 



=-300V 



+— L 



Fig. 9-40 

9-26. Design an astable multivibrator that will generate a pulse whose duration is 
one-tenth that of the total period of 2 msec. Use transistors having /S = 30, a power 



PLATE-GBID AND COLLECTOR-BASE MTJLTIVIBBATOBS 



287 



supply of 10 volts, and two collector resistors of 500 ohms. Make allowance in your 
design for a 20 per cent variation in 0; check to see if recovery is complete before the 
circuit reswitches. 

9-26. In a symmetrical transistor astable multivibrator (Fig. 9-22), the supply 
voltage varies from 10 to 30 volts. R < pR c , where R is the base bias resistor and R e 
is the collector load resistor. Calculate the variation in the period of the multivibrator 
due to the power-supply variation. Explain your answer. 

9-27. The symmetrical astable of Fig. 9-22 employs the following components: 
Ea = 10 volts, Re = 1 K, R = 50 K, C - 1 juf, and = 25. Sketch the base and 
collector waveshapes of one transistor for a complete cycle, labeling them with all 
voltage values, times, and time constants. 

9-28. In the multivibrator of Prob. 9-27, R is reduced to 20 K, a value that ensures 
transistor saturation. Moreover, to guarantee fast recovery, the collector resistance 
is shunted by a Zener diode which will fire at 8 volts (Fig. 9-30). 

(a) Plot the collector and base waveshapes, giving all times and time constants. 

(b) Hot the volt-ampere characteristics of the collector load. 

(c) How will the Zener diode influence the sweep time stability of this circuit? 
9-29. The inductively timed monostable of Fig. 9-27 uses a coil having L = 2henrys 

and Ri = 10 ohms as its basic timing element. As the collector load of the second 
transistor we use a 2-ohm resistor, with Ri = 90 ohms and R t = 200 ohms. The 
power supply is 5 volts, and the switching transistor has /3 = 25 in its active region. 

(o) Calculate the current and voltage waveshapes at the collector of Ti after the 
injection of a trigger. 

(6) Repeat part o when is reduced to 20. 

(e) If the base-to-emitter resistance of the saturated transistor is 2 ohms and the 
collector-to-emitter resistance is 5 ohms, how would the waveshapes of part a be 
modified? 

9-30. Figure 9-41 illustrates an alternative configuration for an inductively timed 
multivibrator. Show that the normal state is with Ti on and T t cut off. Sketch 
the collector and base waveshapes of T 2 and the current in the coil after a trigger is 
applied. Give all times and time constants and the voltage and current coordinates 
at the break points. What are the advantages or disadvantages of this multivibrator 
compared with the circuit of Fig. 9-27? 



1 



+ 10v 





J 50 






;ioo 


Ti 






50 /T> 
-AAV-MT 






Irf 










z 


1,000 
— w\ 





0=25 



Fio. 9-41 

9-31. (a) Derive Eq. (9-37). 

(b) Show that the complete restriction on x must be 



0.5 < x < 1.0 



288 switching [Chap. 9 

(c) For the case of ti = t s , what are the stable states of the two transistors? 

(d) Under the conditions of part c, calculate the loop current gain of the circuit. 
Explain the significance of your answer. 

9-32. The symmetrical multivibrator of Fig. 9-22 is synchronized by injecting 
positive pulses into both bases through diodes. Because of the loading by the on 
transistor, we can assume that these pulses will affect only the off transistor and that 
they will act to shorten the off time. For simplicity, we shall further assume that 
the base charging curve is absolutely linear, taking ti sec to rise from — Ea to zero. 

(a) Calculate and plot the regions of synchronization for n = 1 and n = 2. 

(6) Repeat part o if the pulses are injected into only one base. 



PIBLIOGRAPHY 

Abraham, H., and E. Bloch: Le Multivibrateur, Ann. phys., vol. 12, p. 237, 1919. 

Chance, B., et al.: "Waveforms," Massachusetts Institute of Technology Radiation 
Laboratory Series, vol. 19, McGraw-Hill Book Company, Inc., New York, 1949. 

Clarke, K. K., and M. V. Joyce: "Transistor Circuit Analysis," Addison- Wesley 
Publishing Company, Reading, Mass., in press. 

Eccles, W. H., and F. W. Jordan: A Trigger Relay Utilizing Three Electrode Thermi- 
onic Vacuum Tubes, Radio Rev., vol. 1, no. 3, pp. 143-146, 1919. 

Feinberg, R.: Symmetrical and Asymmetrical Multivibrators, Wireless Eng., vol. 26, 
pp. 153-158, 326-330, 1949. 

Linvill, J. G.: Non-saturating Pulse Circuits Using Two Junction Transistors, Proc. 
IRE, vol. 43, no. 12, pp. 1826-1834, 1954. 

Millman, J., and H. Taub: "Pulse and Digital Circuits," McGraw-Hill Book Com- 
pany, Inc., New York, 1956. 

Puckle, O. S.: "Time Bases," 2d ed., John Wiley & Sons, Inc., New York, 1951. 

Reintjes, J. F., and G. T. Coate: "Principles of Radar," 3d ed., McGraw-Hill Book 
Company, Inc., New York, 1952. 

Toomin, H. : Switching Action of the Eccles-Jordan Trigger Circuit, Rev. Sci. Instr., 
vol. 10, pp. 191-192, June, 1939. 



CHAPTER 10 

EMITTER-COUPLED AND CATHODE-COUPLED 
MULTIVIBRATORS 



10-1. Transistor Emitter-coupled Multivibrator — Monostable Oper- 
ation. The emitter-coupled multivibrator and the equivalent vacuum- 
tube circuit, the cathode-coupled multivibrator, are circuits worthy of 
more than a cursory glance, illustrating asr they do the possibility of one 
circuit operating in any one of several modes. Selection of the particular 
mode and a measure of control over the performance within this mode are 
a function of the control voltage E\, which is labeled in the emitter- 
coupled circuit of Fig. 10-1. 



+30 v 



i\-x)R d 




Ei^xEbb 



Fig. 10-1. Emitter-coupled multivibrator — component values shown for Examples 
10-1 and 10-2. 

Because a general treatment often obscures the detailed functioning of 
the switching circuit, as an introduction, one operating point will be 
explicitly stated and the resultant circuit analyzed. By doing this, a 
straightforward study may be made of the time response without, at the 
same time, being forced to account for the effects of a varying control 
voltage. In Sec. 10-2 the possible modes will be denned and any required 
modification of our original treatment may then be considered. 

We shall assume that the circuit of Fig. 10-1 is adjusted so that, in its 
normal state, Ti is full on and T x is cut off. The application of a positive 
trigger at the base of T x switches the multivibrator into the single quasi- 
stable state: Ti is driven into its active region, and its collector voltage 
drop, coupled through C, turns Tj off. Transistor T lf now operating 

289 



290 



SWITCHING 



[Chap. 10 



as an emitter follower, stabilizes with an emitter-to-ground voltage of 
approximately E%. As C begins charging, the base-to-ground voltage 
of T 2 rises toward Ebb- Once it reaches the new emitter voltage and 
Ci, e 2 equals zero, T 2 will turn back on. The increased current in R 3 
raises the common-emitter voltage above E h and it follows that T\ must 
be cut off. Ed jumps to a higher value, driving Tz back into saturation. 

Note that the coupling from Ti to Ti is through both the common- 
emitter resistance R 3 and the collector base network of C and R. But 
coupling from Ti back to Ti is only through the emitter resistor, which 
leaves R 2 relatively isolated from the transmission path. It therefore 
serves as a convenient place from which to take the output. At this point 
a small amount of capacitive loading may be tolerated without seriously 
affecting the pulse duration. 

In order to simplify the quantitative analysis of the multivibrator of 
Fig. 10-1, we shall make the following further assumptions: 

1. j8 is very large, allowing R S> R x and R S> Ri. 

2. The TheVenin equivalent impedance looking from the base of Ti 
into the control voltage potentiometer is so very small that it also will be 
neglected. 

Preceding switching, the values of all voltages, measured with respect 
to ground, are 

e cl (0") = En (10-la) 

en(O-) = xEn = Ei (10-16) 

R, 



ew(0-) = e c2 (0-) 
e 8 (0-) = E* - 



6.(0-) = 

Rs 



Ri ■+- Rz 



En, = 



Ri + Rz 
Rz 



Eh 



Rz -\- its 



En 



(10-lc) 
(10-ld) 



where e 8 (0~) is the initial charge on the capacitor. 




— fyVV— "° | ( o—WA — 



«! 






Ebb-^- 



(a) (b) 

Fig. 10-2. Emitter-coupled multivibrator — models holding for the quasi-stable state. 
(a) Circuit of Ti\ (6) TheVenin equivalent circuit coupling the collector of Ti to the 
base of T 2 . 

Immediately after switching, the models of Fig. 10-2 represent the 
multivibrator's state and must be used for the calculation of the system's 
time response. From Fig. 10-2o, we can see that the conditions at the 



SEC. 10-1] EMITTER-COUPLED MULTIVIBBATOBS 291 

base and emitter of T\, subsequent to its transition into the unstable 
region, are 

en(0+) = e„(0+) = xE» (10-2o) 

*» (0+) = jfirm (10 " 2&) 

Therefore its collector voltage becomes 

e cX (0+) = E»- faiRt 

where the second term of Eq. (10-3) is simply the drop in collector voltage 
upon switching. Because our original assumption (R 3> RJ allows us to 
ignore the loading of R i by R, the voltage fall at the base of T 2 will be 
identically the drop of Eq. (10-3). We may immediately write 

Recovery from this sharp change of voltage is toward E^, with the time 
constant 

ti = (Ri + R)C £* RC 

Once e it i = 0, or from Eq. (10-2a), once the base-to-ground voltage of 
T t reaches 

€b2(ii) = e, = xEm, 

Ti turns back on. We now know the initial, the final, and the steady- 
state voltage at the base of T^; their substitution into, and the solution 
of, the exponential response equation yields 



Ell— ^ 3 + X ^A 
\ i?2 + Rx Rz J 



h=T ^ Wl-*) " (10 ' 5) 

for the pulse duration. 

The multivibrator reenters the regenerative transition region with the 
base of r 2 at a level lower than its steady-state voltage [Eq. (10-lc)]. 
We suspect that there will be a voltage jump, with the probability that 
this jump will be larger than necessary to return the base to its original 
point. The new base, emitter, and collector voltages can easily be 
calculated by drawing the model holding after the jump, i.e., during 
recovery (Fig. 10-3). 

Just before the multivibrator switches, the final voltage across C is 

e.«i) = E. t - xEn = Ev, \\ - x(l + |fAl (10-6) 



292 switching [Chap. 10 

By substituting this value into Fig. 10-3, we arrive at the new operating 
point of the transistor. The algebra is much simpler when we can use 
numbers, and the solution will therefore be left to the reader as a part 




Fig. 10-3. Recovery model of the emitter-coupled multivibrator. 

20.5 




Fig. 10-4. Emitter-coupled multivibrator waveshapes — monostable operation — mode 
12. Values given are for the solution of Example 10-1. 

of his analysis of specific problems and to the calculations performed in 
Example 10-1. 

After the final jump, the complete circuit recovers toward the original 
steady-state conditions, with the time constant 
r, = C(i?i + Ri || R 3 ) 



Sec. 10-2] emitter-coupled mtiltivibratobs 293 

Waveshapes appear in Fig. 10-4 with all jumps and time constants indi- 
cated. Observe that the base, emitter, and collector of T 2 all have 
identical recovery waveshapes (the saturated transistor is a short circuit). 

Example 10-1. By setting x = 0.4, the monostable multivibrator of Fig. 10-1 is 
adjusted to operate in the mode described above, where T t switches between its 
cutoff and active regions. All circuit components are as given in Fig. 10-1. We do 
not need to specify C for the following analysis, since it would be selected on the basis 
of the required pulse width. 

The duration of the unstable state, which is of primary interest, is controlled by 
the charge in C, maintaining T, cut off. Only two unknown voltages must be found 
in order to evaluate this interval, the initial value at the base of T t as the circuit enters 
its unstable region and the voltage at which T% again turns on. 

From Eq. (10-lc) the stable base voltage of T t is 

800 
e M (0-) = j-g-Q 30 volts = 18.5 volts 

With x = 0.4, the voltage at the base of Ti remains fixed at 

en = En = 12 volts 

When the trigger is injected, Ti turns on and the common-emitter voltage drops from 
18.5 to 12 volts {xE»). 

If the small base current flow is neglected, the voltage drop across ffii will be propor- 
tional to the 12-volt drop across R%: 

jRi _ AEg 
Rl xEhb 

or AE el = 15 volts. C couples this change to base of Tj, driving it from its original 
value of 18.5 down to 3.5 volts. It immediately starts charging, as shown in Fig. 
10-26, toward 30 volts, with the time constant n. When eui = or when en = 
12 volts, Tt turns back on. Consequently 

, ,„ 30 - 3.5 

ti =- ti In _ = O.dSSri 

At the end of the sweep interval the voltage across C is only 3 volts. Substituting 
this value into the model of Fig. 10-3, we find that the base of Ti jumps from 12 to 
20.5 volts. With the time constant 

tj - 1.310C 

the circuit returns to the original conditions. These voltages are used to label the 
typical circuit waveshapes of Fig. 10-4. 

10-2. Modes of Operation of the Emitter-coupled Multivibrator. The 
common-element (emitter) coupling permits this multivibrator to oper- 
ate in a possible multiplicity of modes in each of its two classes. Since 
Ti is always conducting in the normal state, these classes might well be 
defined in terms of the quiescent condition of this transistor: class 1 
existing when the bias current is adjusted so that T% is placed in its 



294 



SWITCHING 



[Chap. 10 



active region, and class 2 when it is saturated. The particular class 
depends solely on the relationship of R to Ri; for T* saturated (class 2) 

R < fiRz 

We can prove the sufficiency of this condition by considering the transis- 
tor model holding under saturation [also see the derivation of Eq. (9-18)]. 

Class 2 operation is to be preferred over class 1 ; here both the initial 
conditions and sweep duration are relatively independent of the transistor 
parameters. However, the saturated transistor has a long recovery 
time and we would be forced to incorporate antibottoming diodes. We 
thus transfer the minority-carrier-storage problem from the transistor 
to the diodes, where it is more readily handled. 

The various possible modes, within each class, are determined by the 
switching conditions of T\, i.e., its starting and ending states. Modes 
will be classified by two numbers, the first describing the state of T\, as 
listed below, and the second the class (state of r 2 ). 

States of T\ 



Before switching After switching 



Cutoff 
Cut off 
Cutoff 
Part on 
Full on 



Cutoff 
Part on 
Full on 
Full on 
Full on 



Since the state of T\ is a direct function of E\, investigation of the 
multivibrator's performance will proceed, following the potentiometer 
setting, from x = to x = 1. Each mode will be delineated by establish- 
ing the boundary values (of x) in terms of the circuit parameters. At any 
particular setting of x, detailed time-response calculations would simply 
follow the analysis outlined in Sec. 10-1. 

We shall mainly concern ourselves with modes n2, representing as 
they do the more stable operation of the emitter-coupled multivibrator. 
In order to complete the discussion one astable mode of group n\ is 
briefly treated in Sec. 10-4. All approximations and assumptions made 
in Sec. 10-1 are also applicable in the following arguments. 

Mode 02. In this mode Ti is normally off. Application of a trigger 
turns T\ part on, but not far enough on so that its collector voltage change 
can drive Tt into cutoff. Upon removal of the trigger, T 2 is still in the 
active region, and it immediately switches back to its normal state, full 
on. The output pulse generated is of small amplitude and of exactly the 
same duration as the input trigger. 



Sec. 10-2] emitter-coupled multivibrators 295 

By assuming that T t has turned full off and that Tx is still part on, the 
model drawn is much simpler than the one required if both transistors are 
taken in their active regions (the model for Tx is given in Fig. 10-2a). 
The region's boundary remains the same regardless of the direction from 
which we approach it. If Tz is cut off, the initial value of its base voltage 
is given by Eq. (10-4). Whenever Tx is on, the drop across the emitter 
resistor is always the control voltage xE^,. In order for T 2 to be part on, 

*- = (ot - '-£)*> - xE » > ° (10 - 7) 

Solving Eq. (10-7) for the limit of x, which satisfies the inequality stated, 
we find that the limits of mode 02 are 

°<*<OTm (10 - 8) 

This calculation serves mainly to define the lower limit of mode 12, the 
first useful range. 

Mode 12. An input trigger will again switch Tx part on, but now the 
drop in its collector voltage is suffi- 
cient to drive T 2 into cutoff. The 
upper limit of x is at that particular "'4> l 

potentiometer setting that would o—Tr E — 

allow Tx to switch from cutoff to I = p I ble 1 bb "■" 

saturation upon being triggered. -=-xE, 
To find this setting, we should draw 



jxE bb B 3 |j09+Ui ( 



the model holding for Tx full on _, ,„,»,,,. tu v„-* r 

, . Fig. 10-5. Model for the upper limit of 

(Fig. 10-5) and solve for the corre- mo de 12. 

sponding emitter-to-ground voltage. 

But for |8 » 1, |8 + 1 = /3, and therefore 

e. = xE» S p ?' p E» (!0- 9 ) 

tix -T «-3 

The limits of x for operation in mode 12 are from the upper limit of 
mode 02 to the value found from Eq. (10-9) : 

Rl E * < x < D R * p (10-10) 



Ri + R s R* + Rs^ Rx + Ri 

For the multivibrator of Example 10-1 the limits are 0.273 < x < 0.445. 

Mode 12 was the one treated in detail in Sec. 10-1. Its pulse width, 

defined by Eq. (10-5), is a function of the potentiometer setting x. 

Increasing the setting increases both the drop in the base voltage of Tj 



296 switching [Chap. 10 

and the value of the emitter voltage after switching into the unstable 
state. As a consequence of the greater separation between the initial and 
final values of the base voltage, the pulse duration increases with potenti- 
ometer rotation. Substituting the limits of x into Eq. (10-5) gives the 
permissible range of output pulse width as 

°< fe <-4 1 + B$7T-i£] (1(M1) 

When the circuit is symmetrical (Ri — R 2 ), the maximum pulse width 
generated is 2i tm « = n In 2. 

Mode 22. In this mode T x switches from cutoff to full on. The 
next possible circuit change occurs when the normal state of Ti changes 
from cutoff to part on. But with T x cut off, Eqs. (10-16) and (10-lc) 
hold: by solving for e he \ = 0, we find the upper limit of mode 22. 

etei = xEn - ^^ ^* E» < (10-12) 

The limits of x for this mode become 



■Rs R 



R 1 + R* <X< RTTR, (1(M3) 

and in Example 10-1, the upper limit is located at 0.615. 

Note that unless Ri > R 2 , mode 22 will not exist and the transition will 
be directly from mode 12 to 32. Furthermore, if R x = R 2 , the range of 
voltage settings for mode 22 operation degenerates into a single point. 

When Ti goes full on, the drop across R 3 is xEu,. The remaining 
voltage must appear across R x ; therefore the drop in the collector volt- 
age of Ti will now be 

Ae„i = (1 - x)E» (10-14) 

This same drop appears at the base of T 2 , turning it off. The base 
charges from its initial voltage, found from Eqs. (10-lc) and (10-14), 

«m(0+) = jjpjrij; E»-(l- x)E vb (10-15) 

toward Ett, with the time constant n. It finally conducts when 

e«(<2) = xEu, 
Consequently, in mode 22, the pulse width becomes 

h = r t In 2 ~ Ri/ ^l\ Ri) ~ X (10-16) 



Sec. 10-2] emitter-coupled multivibkatoks 297 

Mode 32. Under the circuit conditions existing within this mode of 
operation, Ti always conducts, maintaining the emitter voltage fixed 
at xEn, across the complete transition region. Limits, on x, are from 
the upper limit of mode 22 to that 



value which puts both transistors i 1 

into permanent saturation (Fig. flj s Rl S 

10-6). f f 

The upper limit, found from the _[] X £».-=■ 

model of Fig. 10-6, is -=-xE bb <# 3 

R 

xEu < ■= — ,, D 8 — 5- En, Fiq. 10-6. Both transistors in saturation. 

Ri II «2 + R% 

Thus, in mode 32, the range of x becomes 

Ri ^ - R% ,-- 1 _i 

Rz + R* <X< Rx || R* + R, {1 °- 17) 

When R 3 is very large compared with Ri and R 2 , the limits of x for which 
this mode holds shrink almost to the vanishing point. The circuit 
conditions before switching are determined from the appropriate model 
(Ti part on and T 2 saturated) : 

en(O-) = e» 2 (0-) = e.(O-) = xEn (10-18a) 

«.i(0-) = E» - E» [^ - (1 ~ R f Rl ] (10-186) 
But after switching, 

eci(0+) = xEu, (10-19) 

The net change in the collector voltage of 7\ [Eqs. (10-186) and (10-19)] 
is coupled to the base of T^, and as the multivibrator enters the quasi- 
stable region, Ae c i determines the initial value of e b2 : 

e*m = (2x - 1)E» + [^ - (1 ~ R f Rl ] V* (10-20) 

As in all the other modes, the base charges toward En, switching at xEu,. 
The pulse duration, still a function of x, turns out to be 

t. = ri In *' \_* R *> (10.21) 

In the three modes of operation that generate a usable pulse, its dura- 
tion, at least to the first approximation, is independent of £», resulting 
in extremely good pulse stability. 



298 switching [Chap. 10 

Mode 42. This mode exists when both transistors are always satu- 
rated. Thus an input trigger will have absolutely no effect. The limits 
are 

Rz 



Ri II Ri + Rz 



< x < 1 



(10-22) 



It is included only to complete the range of setting of the potentiometer 
and, as mode 02, is of no practical use. 

A summary of the results is presented in Tables 10-1 and 10-2. Table 
10-1 shows the operating regions. Within each mode the normalized 
pulse duration is given by 

b + ax 



tn 



T 1 — X 



(10-23) 



The constants a and b are listed in Table 10-2. 

Table 10-1. Transistor States, Emitter-coupled Multivibrator 



Mode 


Upper limit of x 


T l 


T 2 


02 


Rt Rs 


Off 


Full on 


Ri 4- Rt R2 + Rt 


12 


R, 


Off 

I 
part on 


Full on 

i 
cut off 


Ri + Rt 


22 


R, 


Off 

1 
full on 


Full on 

1 
cut off 


R% 4~ Rz 


32 


R, 


Part on 

1 
full on 


Full on 

I 
cut off 


Ri 1! ^2 + Rz 


42 


l 


Full on 


Full on 



To summarize further the behavior of the multivibrator in terms of 
the particular modes, the complete normalized period is plotted as a 
function of x for the circuit of Example 10-1. The solid curve 1 of 
Fig. 10-7 illustrates the varying pulse duration as a function of x when 
the impedance of the control potentiometer is negligible. This cor- 
responds to the mode boundaries discussed above. Both the demarca- 
tion between the individual modes and the different rates of pulse-width 
variation with x within each mode are quite clear. It might be noted 
that, in this special case, the pulse duration appears to vary almost 
linearly with x within mode 12. 



Sec. 10-2] 



EMITTER-COUPLED MTJLTIVIBRATOBS 



299 



Table 10-2. Pulse Duration or Emitter-coupled Multivibrator — 

Modes n2 



Mode 


6* 


a* 


12 


1 R ' 


Ri 


Rz "h R3 


R, 


22 


R> 


-1 


Rt -f- R$ 


32 


Ri 
2+ R, 


\ Rt Rt) 



* Constants for Eq. (10-23). 

Effects of Finite Potentiometer Impedance. When the control potentiom- 
eter is of appreciable size, we must include its equivalent source imped- 
ance in all calculations involving the actual base and emitter voltage of Ti. 
The dashed curve 2 of Fig. 10-7 shows the effect of a 10-K potentiometer 

0.8 

0.7 

0.6 
i 0.5 
!s 0.4 

T 

0.3 
0.2 
0.1 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 L0 
Fig. 10-7. Pulse duration as a function of potentiometer setting. 

on the pulse duration in the various modes of operation. This plot 
is the result of inserting the Thevenin equivalent impedance of 

R. = x(l - x)R d (10-24) 

in series with xEn. 

For mode 12, the input model is shown in Fig. 10-8a. Here, for any- 
setting of x, the source impedance is much less than the active-region 
input impedance (/3 + 1)R 3 of the transistor. With the specific circuit 
values of Example 10-1, the input impedance is 56 K compared with the 



^ 








































-fl 2 -500 
-#3=800 
-/3-70 
-B-25K 

- £,,6=30 v 


































Mode 22- 




























































1 


























*• 


"? 




\ 
















-.( 


) _^ 




/ 








\ 






loc 












n v j 


'/ 








\ 














/ 






































■/, 












T 


















_Mode 12. 


J 


II 












ft 




























1 






































It 








































II 




















































e< 












_J 






II 


















L 












300 switching [Chap. 10 

maximum R a value of 2.5 K. The loading of the potentiometer is 
negligible, and the base voltage is but slightly less than the open-circuit 
voltage xEw,. To generate a given-duration pulse, the value of x must be 
set slightly higher than previously found, in order to compensate for the 
voltage drop across R,. 

Once the multivibrator begins operating in mode 22, the input model 
must be changed to the one shown in Fig. 10-86, which is obtained 
by replacing the transistor by the Thevenin equivalent circuit seen 
looking into the saturated base. The input impedance is reduced 
from 56 K to the parallel combination of Ri and R 3 (445 ohms). Fur- 
thermore, the potentiometer is varying about the mid-point of its 
setting, and its impedance remains close to the maximum possible value 
{R, = 2,500 ohms). Because most of the voltage will be developed 
across R„ any change in the control-voltage setting will have but small 

x(X-x)r d x(l-x)r d Cj R^Ri 

J_ i»i t e i 445 _j_ 

(b) 

Fig. 10-8. Equivalent input circuit of T u including the potentiometer source imped- 
ance, (a) Mode 12 operation; (6) Ti saturated (mode 22). 

effect on the base voltage of TV The common-emitter voltage will not 
vary much, and neither will the pulse duration. This segment of the 
curve in Fig. 10-7 becomes much more horizontal. In Example 10-2 
we shall consider one specific setting of x, lying in this mode, to illustrate 
the role of R„ in determining the circuit voltages and timing. 

Mode 32 operation is restricted to such a small fraction of the range 
that, in general, it is of little interest. R, would be treated as above, by 
including it in the appropriate models. The reader may perform this 
substitution and calculation himself. 

Example 10-2. The multivibrator of Fig. 10-1 uses a 10-K potentiometer for its 
timing control. When it is set at x = 0.55 the circuit is operating near the center of 
mode 22. All other circuit components are the same as in Example 10-1, and the 
results of that problem may be used in this solution. 

The initial voltage at the base of T 2 , before triggering, is 18.5 volts (from Example 
10-1). After the trigger turns 7\ on, the saturated transistor voltage may be found 
from the model of Fig. 10-86. Here xEu, = 16.5 volts, and R, = 2,475 ohms. From 
Fig. 10-86, 

eei -»,-«.- 13.33 + ^~ X (16.5 - 13.33) 

= 13.81 volts 

If R, were 0, this voltage would be xEu, = 16.5 volts. The loading of the potentiom- 
eter reduces the transistor voltage to slightly above its saturation value. Thus, as 7\ 



Sec. 10-3] emitter-coupled mtjltivibbatoks 301 

turns on, the drop at its colleetor is only 

A« el = 30 - 13.81 = 16.2 volts 

This drop is coupled by C, driving the base of Tt down to 

en(0+) = 18.5 - 16.2 = 2.3 volts 

Recovery is toward 30 volts, with the circuit reswitching at the common-emitter 
voltage of 13.8 volts: 

When the source impedance is small enough to be neglected, the voltage at the 
terminals of 7\ will be xEu (16.5 volts). In this case, the change in collector voltage 
is only 13.5 volts and the base of T% is driven to 5 volts. The reswitching value is also 
higher, 16.5 volts, and since a greater percentage of the total exponential is used, the 
sweep period becomes somewhat longer: 

*'- Tim 30°-~16.5 = - 615r ' 

10-3. Monostable Pulse Variation. The question confronting us in 
this section is, What circuit relationships must be satisfied to achieve 
some physically permissible variation of the pulse duration with respect 
to x? In modes 12 and 32, the function might well be a linear one, 
and in mode 22 it might be either constant or linear. One commonly 
used method of attacking this problem is first to expand the general time 
equation (10-23) into some series and then, by examining the individual 
terms, attempt to eliminate those that contribute the nonlinearity. 
But since x takes on a relatively large range of values, there is no possible 
series that will converge fast enough so that all the higher-order terms 
in x can be eliminated. A solution of this type is quite tedious, and in 
addition, the design requirements are often obscured beneath the algebra 
needed in handling the existent higher-order terms. 

A second, more fruitful approach is through an investigation of the 
properties of the slope of the curve rather than of the curve itself. When 
we differentiate Eq. (10-23) with respect to x, the resultant expression 
for the slope is no longer an exponential function of x, and consequently 
it is of simpler form than the original equation : 

S = M . -(» + .) (10 _ 25) 

ax ax 2 — (a — b)x — b v 

In order for the pulse-width variation with respect to changes in x to be 
linear, the slope must be independent of a;; it should be a constant. But 
this requires that the denominator of Eq. (10-25), 

y = ox 2 - (a - b)x - b (10-26) 



302 switching [Chap. 10 

must also be a constant for all values of x within any one mode. Equa- 
tion (10-26) is the equation of a parabola, and therefore it is impossible 
to satisfy the required condition; we can never expect a completely linear 
relationship. A good compromise will keep the change in y small for 
the range of x within the mode under investigation. 

We would conclude, from scrutinizing the properties of a parabola, 
that it is most nearly constant with respect to small variations of x in 
the vicinity of its vertex. The best design would adjust the circuit 
parameters so that the point of inflection of Eq. (10-26) lies in the middle 
of the mode in which the linear function is desired. Differentiating 
Eq. (10-26) and equating to zero locates the vertex at 

For mode 12 of the multivibrator that was treated in making the plot of 
Fig. 10-7 (zero control-source impedance), x mi is at 0.345 compared with 
the lower and higher limits of Xn = 0.272 and Xm = 0.444. It is quite 
close to the optimum placement of the vertex at x = 0.358. But in 
mode 22 of the same circuit x m % is at 1.19, well outside the mode limits of 
xn = 0.444 and Xk% = 0.614. The difference in linearity is quite apparent. 
The second condition for good linearity also comes from the examina- 
tion of the properties of the parabola. If the focus is close to the vertex, 
then the parabola is wide open and will be more nearly constant in the 
vicinity of the vertex, further improving the linearity. This distance, 
for the parabola of Eq. (10-26), is 

4a 

Generally it is not possible to make D small and still satisfy Eq. (10-27). 
Setting D = in mode 12 is a trivial case, requiring a zero value for R it 
which would make the multivibrator inoperative. But even in this case, 
setting R 3 as small as possible compared with Ri minimizes D within the 
other design constraints that now determine the lower limit of .fi^- The 
more important condition is the one given by Eq. (10-27), and this is the 
one which must be satisfied for best linearity. 

Under some other circumstances it may be desirable to make the pulse 
width relatively independent of x. From Eq. (10-25) it can be seen that 
this requires a = — b. All modes give ridiculous answers in satisfying 
this condition, either a pulse duration of zero or negative values of circuit 
resistors. 

By including the source impedance of the control path in the deriva- 
tion of the timing waveshape, we would find it possible to maintain the 
pulse width almost constant with respect to small changes in a; in mode 22. 



Sec. 10-4] 



EMITTER-COUPLED MULTIVIBRATORS 



303 



The equation resulting would be in the same form as Eq. (10-23), with, 
of course, different constants. Either from this equation or by consider- 
ing the effect on the base voltage of Tx (as in Example 10-2), the slope 
of the time-versus-x curve can be minimized by maximizing R,. Any 
changes in the transistor parameters would now have but little influence 
on the timing, and the sweep stability would be greatly improved. 

10-4. Emitter-coupled Astable Multivibrator. Astable operation, at 
least according to Sec. 9-7, apparently is dependent upon two independent 
energy-storage elements maintaining first one and then the other tran- 
sistor cut off. But in the circuit configuration of Fig. 10-1 or 10-9, a 
single capacitor controls the duration of both unstable states. To see 
how, suppose that, with the feedback loop broken and by satisfying the 
following conditions, we bias both transistors in their active region. 

For r 2 active: 

R > fiRz 
For Tt active: 



(1 + g)fl 8 
R + Q.+ fi)R, 



< x < 



R 3 R + (1 + fi)RzRi 
R(Rt + R 3 ) + (1 + /»)«,«! 



(10-28) 



+25 v 



The complete limits of astable operation expressed by Eq. (10-28) 
are readily derived from the appropriate models, one with Ti cut off and 
5" 2 active, and the other for Ti saturated while T 2 remains active. 

Once the feedback loop is again closed, the regenerative action of the 
multivibrator rapidly drives T 2 into cutoff. As C recovers toward 2J», 
the base of T 3 eventually reaches the 
emitter voltage xEu,, at which point 
T2 reenters its active region. The 
increased current through R t now 
raises e, above xEu, and thus drives 
Ti off. The resultant jump in the 
collector voltage of Ti forces T 2 to- 
ward, or even into, full conduction, 
capacitor charging current contrib- 
uting the necessary additional base 
current flow. However, C will again 
recover. In the process, the total 
emitter current drops; the base voltage of T 2 and the common-emitter 
voltage exponentially decay, finally recrossing the xEu line. At this 
instant T t turns back on, and the cycle repeats. We see that the single 
capacitor performs both timing functions; first it maintains T 2 cut off, and 
next it supplies the additional current that keeps T 2 turned on and/or 
Ti cut off. 




Fro. 10-9. An astable emitter-coupled 
multivibrator. 



304 



SWITCHING 



[Chap. 10 



As a convenient starting point for the purpose of computation, we 
might well look at the conditions of Fig. 10-9 while T 2 is on, just going off, 
and Ti is off, on the verge of turning on. The initial voltages at t = Qr, 
neglecting any charging current which might be flowing, are 



e c i = E» — 25 volts 

e&i = e& 2 = e, = xE bb = 7.5 volts 



Gq — €el 



e&2 = 17-5 volts 



Immediately after the circuit changes state, at t = + , T 2 becomes cut 
off and Ti active. Therefore the model with which we must be concerned 
is the one shown in Fig. 10-10a. Since the 100-K base bias resistor of 
!T 2 is so very much larger than the 1-K collector resistor of T\, we may 




?+25 



•IK 



100K 



333 



*3(l + /3)< 
=■25 K< 



(c) 



Fig. 10-10. Transistor models used for the calculation of astable operation, (a) 7\ 
on, T 2 off; (6) T^ off, T 2 saturated; (c) Ti off, T 2 active. 

ignore the additional loading introduced by R. From Fig. 10-10a, the 
conditions at t = 0+ are 



en = e, = 7.5 volts 

7.5 volts 
%b\ = 



= 0.3 ma 



(50 + 1)500 
e cl = E» - 504i (1 K) = 25 



15 = 10 volts 



The collector voltage falls 15 volts, and the base of T 2 drops by the same 
amount, from 7.5 to —7.5 volts. It immediately starts charging toward 
En with the time constant 

tx = RC = 20 msec 

But Tz turns back on once e 62 = e. = 7.5 volts, and thus the first portion 
of the cycle lasts for 

(l = Tl ln |»J=-5* = 20 In 2b ~ ( ~ 7 ; 5) = 12.36 msec 



En 



E, 



25 - 7.5 



Sec. 10-4] emitter-coupled multivibrators 305 

The final charge in C, which we shall need as a boundary condition in the 
next operating region, is 

e«('i) = e«si(<i) — e t2 (<i) = 2.5 volts 

After Ti turns back on, the jump in the collector voltage of T x as it 
turns off, coupled through C, forces T 2 into saturation (Fig. 10-106). 
Before we can calculate the time response, we must verify the correct- 
ness of this assumption. From the model drawn for the saturated tran- 
sistor, we see that the equivalent-emitter voltage is 

ElbRz 8.33 volts 



R\ + #3 



The additional capacitor charging current flowing through the 333-ohm 
source impedance (R* || Ri) raises the voltage of all the elements of T t 
to 11.87 volts, a value well above the saturation-threshold voltage. The 
initial current flow through C and the base of T» becomes 

. ». 25 - 2.5 - 8.33 , na 
»m(U) = ■ = 10.6 ma 

However, the base current necessary to sustain saturation is merely 

*' - (/J + l)fi, = 25K = °- 333 ma 

and of this, the amount contributed by the normal bias current flow 
through R is 

Eg - e, 25-8.33 

*» = — r - = ioo k = 0167 ma 

Thus the equation of the base current component due to the capacitor 
charging current becomes 

iuc(t) = 10.6e-"" 
where T2 = (fl t + R 2 1| R 3 )C = 0.266 msec 

T% finally enters its active region (from saturation) when 

ibtcih) = 0.167 ma 

As a consequence of the large initial and small final value of current flow, 
the duration of the first portion of the recovery period takes virtually 
the complete exponential: 

1 ft R 

h = Tt In Q^Tjy = 4r 2 = 1.06 msec 

At the end of this interval, a new model is needed for r 2 , one represent- 
ing the active region (Fig. 10-10c). The transistor input resistance, 



306 



SWITCHING 



[Chap. 10 



measured from base of 7% to ground, is (0 + 1)22, = 25 K, and the 
equivalent input open-circuit voltage is 

(0 + 1)« 3 „ 25 K 



En — 



R + G8 + 1)«; 



•Em, = 



100 K + 25 K 



25 volts = 5 volts 



Moreover, since the transition from the saturation to the active region 
will be smooth (without any voltage jump), there is no need to calculate 
the initial charge on C. Charging continues, but now toward 5 volts, 
with a new time constant t 3 = [(j3 + 1)R 3 \\ R]C = 6.7 msec. Once the 
common-emitter voltage falls to 7.5 volts, Ti again conducts, the circuit 
reswitches, and the cycle repeats: 



U = t 3 In 



5 - 8.33 
5 - 7.5 



= 1.94 msec 



During this final interval Tt is in its active rather than its saturated 
state. As its base current continues to decrease, the collector voltage 
now starts rising. The generated waveshapes at the base of T 3 and at 
the common-emitter junction appear in Fig. 10-11. 




Fig. 10-11. Controlling waveshapes of emitter-coupled astable multivibrator. 



Sec. 10-5] 



EMITTER-COUPLED MULTIVIBRATORS 



307 



10-6. Cathode-coupled Monostable Multivibrator. A vacuum-tube 
cathode-coupled circuit (Fig. 10-12) functions in a manner quite similar 
to that of the common-emitter transistor multivibrator; the principal 
differences involve the circumstances that surround the change of circuit 
state. These naturally arise from 
the physical properties of the vac- 
uum tube and will be represented 
in the circuit models. The various 
operating modes would be defined 
for the same switching conditions 
as in the transistor circuit. 

All aspects of this multivibrator's 
operation in mode 12, the only one 
with which we shall be concerned, 
are presented in the two models 
drawn below. The first represents 
the circuit conditions both during 




Fiq. 10-12. Cathode-coupled monostable 
multivibrator. 



recovery and while the multivibrator is in its normal state (Fig. 10-13a), 
and the second, when it is in its quasi-stable state (Fig. 10-136). 

In an effort to simplify computations, we shall make the following 
assumptions: 

1. The equivalent grid resistance in the positive grid region, r„ is 
small compared with R. 

2. R is sufficiently large so that its loading of Ri may be neglected. 

3. The contribution of grid current in developing the voltage across 
Ri is insignificant. 

From Fig. 10-13a (T t off, T % saturated), the quiescent circuit voltages 
are , 



e 5t 2(0-) S 
e ci (0~) S e*(0-) 



#3 



R% + #3 + r p 
r P + R 3 



Eu 



e g (0~) =• Em- e c2 (0~) = 



Eu 



r P + R* 



R2 + R% + T t 



Eh, 



(10-29o) 
(10-296) 

(10-29c) 

(10-29d) 



For proper operation in this mode, T\ must be cut off. The required 
minimum bias voltage (grid to cathode) is determined by the plate-to- 
cathode drop. 



Eta\ — — 



Ey, 



En 



et 



(10-30) 



308 switching [Chap. 10 

and to ensure that T\ remains off in its normal state, 

»tki = xE» - et < - Ebb ~ e " (10-31) 

where e k — RzEti,/(rp + R 2 + R 3 ). 

As the multivibrator enters upon its quasi-stable state, after the appli- 
cation of an external trigger, Ti switches from cutoff to its active region. 
The plate and cathode voltages become (Fig. 10-136) 



e*(0+) = 



r P + Ri+ (m + 1)R, 



«n(0+) = E»- 



r P + Ri+ <jt + 1)B, 



(10-32) 
(10-33) 



We must check that e*(0+) > xE^, because if this inequality is not 
satisfied, the model of Fig. 10-136 is no longer valid. It must then be 




Fig. 10-13. Cathode-coupled multivibrator circuit models, (o) Normal state; (6) 
quasi-stable state. 

replaced by the model representing positive grid operation. The second 
term of Eq. (10-33), the drop in plate voltage upon switching (Ae»i), 
is coupled to the grid of T-t, driving it well below cutoff: 



«c S (0+) = e c2 (0-) - he n 
RtEvb 



Rx(\ + ;js)-Ew 



Ri + Ri + r p r P + Ri + (n + 1)2?, 



(10-34) 



The grid of T t charges from its initial value [Eq. (10-34)] toward En, 
finally switching at t — h as it reaches its particular cutoff voltage: 



e„*(ii) = e*(0+) - 



En, - e>,(0+) 



(10-35) 



Sec. 10-5] emitter-coupled multivibrators 309 

where e*(0+) was given in Eq. (10-32). And for high-/* tubes, where 
( M + 1)R, » r, +• R u 

The charge time constant, again found from Fig. 10-13&, is 

r, = C{« + R t || [r, + ie 3 (M + 1)]} £< «C 
and thus the pulse duration becomes 

"-""ttSB < 10 - 36 > 

If we substitute the two values of e c2 found above into Eq. (10-36), 
we shall see that the pulse width is independent of E a and, in addition, 
that the resultant equation is of the same form as the general timing 
equation (10-23). For the particular case of a high-ju tube, we obtain the 
equation 

i R* , 1 + nx Ri 

*i-nln Ri + R \+_ r ' x *— *« (10-37) 

At the end of the unstable state, the voltage across C is 

e,(*i) = e»i(0+) - ea(h) 

and by substituting this value into the circuit of Fig. 10-13a, we can 
compute the initial value of the positive grid excursion as the multi- 
vibrator starts recovering toward its normal state. Recovery at the 
plate of Ti and the grid of T 2 is with the time constant 

T 2 = C[fi, + r./(l - A k )] 

where the positive gain from grid to cathode effectively multiplies the 
grid resistance (A k < 1). We see, at both the plate and cathode of T 2 , the 
reflected grid recovery waveshape (Fig. 10-14), amplified by the appropri- 
ate factor. Waveshapes of all tube elements, further illustrating the 
performance of this circuit, appear in Fig. 10-14. 

Limits of the various operating modes of the vacuum-tube multi- 
vibrator are computed from the appropriate models by simply solving 
for their boundary values. Since, in its active region, the vacuum tube 
operates within a grid base, dependent not only upon the voltage across 
the tube but also on the tube parameters, the factors defining the limits 
become quite unwieldy. In the special case of the high-/i tube, there 
will be considerable simplification. However, for any specific multi- 
vibrator, where subcalculations using the circuit resistance and the tube 



310 



SWITCHING 



[Chap. 10 



parameters are usually performed, the problem is straightforward and 
not at all difficult. 

10-6. Limitation of Analysis. Physical multivibrators differ in several 
important aspects from the idealized circuits treated in this chapter. 
Experimentally measured results cannot be expected to agree exactly 
with those calculated on the basis of our previous discussion. Since 
we now understand the multivibrator's operation, we might take a 




t, t 

Fig. 10-14. Cathode-coupled multivibrator waveshapes — monostable operation. 

second look at the terms previously ignored, with a view toward finding 
out how they would modify the original analysis. 

One of the two terms affecting the pulse duration of the transistor 
multivibrator is the small voltage drop, about 0.1 or 0.2 volt, that 
actually appears between the base and emitter of the conducting tran- 
sistor. This voltage could be included in the transistor models by insert- 
ing a small battery in the base emitter circuit. If the transistor switches 
to well below cutoff and starts charging toward E», it seems completely 
reasonable to neglect 0.1 volt compared with the total base voltage change 



Sec. 10-6] emitter-coupled multivibrators 311 

of 5 to 20 volts or more. However, near the lower limit of mode 12 and 
the upper limit of mode 32, the transistor is driven only slightly below 
cutoff and the resultant error in the pulse duration will be quite large. 
The generated pulse will always be shorter than the calculated pulse, thus 
introducing additional curvature in these regions of Fig. 10-7. 

The second factor, which also tends to reduce the pulse duration, is 
the temperature-dependent /„<>. This, the reverse collector-to-base 
current, contributes an additional charging component to C while T% 
remains cut off. And of course faster charging means that the pulse 
will be shorter than expected. I c0 may be 5 /ta or less at 20 9 C, but it 
increases exponentially with temperature, even reaching 50 to 100 n& 
at the elevated ambient temperatures often encountered. As long as the 
normal charging current through R remains much larger than I c0 , the 
temperature effects are minimized. Thus another reason appears for not 
operating under modes nl, where R is, of necessity, quite large. Further- 
more, some form of temperature compensation is usually employed in 
transistor circuits. 

The extreme curvature of the plate characteristics at low values of 
plate current leads to the tube's turning on at a lower grid voltage than 
that given by the piecewise-linear model. Consequently the charging 
exponential is interrupted sooner and the generated pulse will always be 
shorter than calculated. A larger plate drop is necessary to ensure driv- 
ing the other tube into cutoff, and therefore the lower limit of mode 12 
will occur at a higher setting of x and the upper limit of mode 32 at a 
lower setting of x. The difference from the ideal would be appreciable 
for 1ow-m tubes but would become negligible when high-/* tubes are 
employed. 

The second-order effects are primarily discussed to show that where 
physical and theoretical results disagree, a closer look at our original 
assumptions, or at the active elements themselves, will often explain the 
source of discrepancy. Practical circuits always have adjustments for 
timing. Since 5 or 10 per cent accurate components and 20 to 50 per cent 
tolerance in tube and transistor parameters are what the designer must 
cope with, exact design is neither possible nor desirable. Thus a rapid, 
simple, approximate treatment is often more satisfactory than an exact 
analysis, provided that the answers obtained are reasonable. 



PROBLEMS 

10-1. The emitter-coupled circuit of Fig. 10-1 employs the following components: 
fl, - R t = 2 K, R, = 1 K, R = 50 K, C = 1 »f, - 50, and E» = 5 volts. 

(a) For xEu, = 0.75 volt, plot to scale the waveshapes seen at both collectors, at 
the base of T t , and at the common emitter. 



312 



SWITCHING 



[Chap. 10 




Fio. 10-15 



(6) Repeat part a for xEu, «■ 1.5 volts. Superimpose these plots on the wave- 
shapes drawn for part a. 

10-2. We wish to adjust the multivibrator of Prob. 10-1 so that it will generate a 
2-volt 10-msec pulse at the collector of T t . This should be the maximum possible 
pulse width which can be produced by this circuit. 

(a) Specify the new values of R», x, and C needed to satisfy these conditions. 
(6) Sketch the waveshape appearing at the base of TV 

10-3. (a) Calculate the limits on x, 
+ 10v separating the various modes of opera- 

tion for the multivibrator of Fig. 10-15. 
Select a value of x that lies halfway 
between the extremes of mode 12, and 
calculate the pulse duration and ampli- 
tude at the collector of T% (assume Rt 
small). 

(6) Solve for the setting of x that will 
produce a pulse of the same duration but 
in mode 32. Plot the collector wave- 
shape to scale on the same graph as in 
part a. 

10-4. Design an emitter-coupled multi- 
vibrator that will generate at its maximum 
setting, a 10-msec pulse. This circuit should switch directly from mode 12 to mode 32 
and must operate in mode 12 over the widest possible range of x. Base your design 
on a transistor having /8 = 50 and E» = 5 volts. Set Rt = 1 K. Specify the range 
of x for both operating modes. Sketch the various circuit waveshapes produced when 
the multivibrator generates its 10-msec pulse. 

10-5. (a) The multivibrator of Fig. 10-15 is controlled by a 6-volt battery in place 
of the potentiometer. Ett suddenly drops by 5 per cent, from 10 to 9.5 volts. What 
effect will this have on the pulse duration? 

(6) If this same control voltage is derived by setting the potentiometer at 
x = 0.6, by how much will the pulse width change when En is reduced by 5 per 
cent? 

(c) Repeat parts o and 6 if a 2,000-ohm resistor is inserted in series with the base 
of T t . 

10-6. (a) Show three methods of triggering the multivibrator of Fig. 10-15. Dis- 
cuss the required source impedance of the signal generator and the means of coupling 
the pulse into the circuit. 

(6) If a pulse is applied through a diode to .the common-emitter terminal, calculate 
the required amplitude at x — 0.2, 0.4, 0.6. 

10-7. (o) The emitter-coupled multivibrator shown in Fig. 10-15 must be adjusted, 
by changing Ri, to generate the longest possible pulse. If x = 0.4, find the required 
value of fli and the width of the pulse produced. 

(6) With Ri = 1 K and with x set to give the maximum possible swing at the collec- 
tor of Ti, sketch and label fully the waveshape at the collector of Ti. 

(c) Sketch and label the waveshapes at both collectors when x = 0.33 and when 
/Si = 1 K. 

10-8. (o) Plot the collector voltage swing of 7: as a function of the potentiometer 
setting for the multivibrator described in the text in Example 10-1. Label all modes 
of operation. 

(6) Repeat the plot, on the same graph, for the collector voltage of Tu 



14 



Shaft 



EMITTER-COUPLED MULTIVIBRATORS 313 

10-9. The multivibrator whose characteristics are plotted in curve 1 of Fig. 10-7 
is used in a special instrument to measure small linear displacements. This informa- 
tion is contained in the width of the pulse produced once the multivibrator is triggered 
with a read-out puke. For the transducer we employ a small capacity, replacing C 
in Fig. 10-1, whose effective plate spacing is varied by the v ,. , 
shaft displacement (Fig. 10-16). Assume that the minimum ^ I 
value of C is 1,000 npi for d = 10 -4 cm and that its maximum % ' 
value is 10,000 prf for d = 10~ s cm. Moreover, to prevent % 
arc-over, Eu is reduced to 2 volts. % 

(a) When x =0.3, calculate the range of pulse widths % 
generated as the rod is displaced. M 

(6) An angular rotation $ of ±3° varies x between the „ „ , 

limits of 0.2 to 0.4 from its nominal value of 0.3. Under 
these conditions, express the pulse duration < as a function of d and 0. 

t, =f(d,B) 

Evaluate all constants. Assume a linear variation of h with respect to both d and 8. 
Is this assumption justified? 

10-10. (a) Verify the bounds of each mode for the multivibrator discussed in the 
text in Example 10-2. Assume that R, remains constant at 2,500 ohms. Is this 
approximation justified? 

(6) Prove that the x boundary between mode 22 and mode 32 is independent of the 
potentiometer employed for control. 

10-11. In the circuit of Fig. 10-15, R d is a 5,000-ohm control. Plot the waveshapes 
at each transistor element, to scale, when x = 0.5. How does the pulse duration com- 
pare with the one generated when R d is very small? 

10-12. The equation 

y = In ^- < x < 0.75 

1 — x 

defines the operating path of a circuit. 

(a) For what value of x is y most nearly a linear function of x? 

(b) If x is varied by ±0.05 about this point, what is the nonlinearity of y? 

(c) What is the NL of y if x varies by ±0.05 about the point x = 14? 

(Define the NL of y as the maximum variation in y from a straight line drawn between 
its bounded end points divided by the change in y between the same two end points.) 

10-13. (a) Derive the limits of Eq. (10-28). 

(6) Find the numerical values of x that delineate the modes of operation of the 
multivibrator discussed in Sec. 10-4. Tabulate, under each mode, the normal and 
the switched state of each transistor. Are any modes of operation monostable? 
Explain. 

10-14. Repeat the calculations for the sample problem given in Sec. 10-4 when x is 
increased to 0.5. Make all reasonable approximations. 

10-16. Determine the amplitude and duration of the output taken at the plate of T* 
in Fig. 10-12 if x = H, Ri = fi 2 = 10 K, R, = 5 K, R = 500 K, C = 1,000 M, 
r p = 20 K, fi. = 40, and En = 300 volts. 

10-16. (o) Find the minimum and maximum values of E t that still permit the 
circuit of Fig. 10-17 to function as a monostable multivibrator. 

(b) With Ei adjusted to 5 volts below its maximum point, sketch and label the 
waveshapes produced at et u ew, e t , and e„i when a trigger is applied at t = 0. 



314 



SWITCHING 



[Chap. 10 



?+300 



10 K 



Hf- 



10 K 



(c) In which modes would the source impedance of Ei affect the pulse duration? 
Explain your answer. 

10-17. The circuit of Fig. 10-17 is modi- 
fied by returning the timing resistor to 
ground instead of to En and by increasing 
the capacity to 2 /if. Solve for the response 
at all pertinent elements after a trigger is 
injected, if Ei = 50 volts. Does this 

I ^ i > h i modification improve or degrade the opera- 

[ + V3 Y ^— f tion? Explain. 

-^-JBj I i 1 10-18. The transistor multivibrator of 

Ir=10K Sck ^ s " i"" 1 ^ ' s designed to operate at very 

/«=20 ? low voltages. Consequently its small 

_ re- 500 -i- emitter base drop must be taken into 

account in calculating the time duration. 
Furthermore, the temperature-dependent 
ho increases from 1 ^a at 20°C to 20 /ta at 100°C. Calculate the nominal pulse 
duration and amplitude at the collector of T% (at 0°C). By how much will the pulse 
vary over the expected temperature range ? For your calculations use the transistor 
model of Fig. 10-186. 




0.01 nX 



r p =10K 

A=20 

rc-500 

Fig. 10-17 




+1.5 v 




bo- 



03+U/co 

— o— 

-* 0'L 



:=:o.2v 



T- 



(b) 



Fig. 10-18 



BIBLIOGRAPHY 



Chance, B., M. H. Johnson, and R. H. Phillip: Precision Delay Multivibrator for 
Range Measurements, MIT Rod. Labs. Rept. 63-2. 

Clarke, K. K., and M. V. Joyce: "Transistor Circuit Analysis," Addison-Wesley 
Publishing Company, Reading, Mass., in press. 

Glegg, K. : Cathode-coupled Multivibrator Operation, Proc. IRE, vol. 38, no. 6, pp. 
655-657, 1950. 

Millman, J., and H. Taub: "Pulse and Digital Circuits," McGraw-Hill Book Com- 
pany, Inc., New York, 1956. 

Reintjes, J. F., and G. T. Coate: "Principles of Radar," 3d ed., McGraw-Hill Book 
Company, Inc., New York, 1952. 

Schmitt, O. H.: A Thermionic Trigger, J. Sci. Instr., vol. 15, p. 24, 1938. 



CHAPTER 11 

NEGATIVE-RESISTANCE SWITCHING CIRCUITS 



A two-terminal active network that exhibits a negative-resistance 
driving-point impedance over a portion of its range may, together with 
an energy-storage element, be used as a switching circuit. Our initial 
discussion concerns a postulated ideal device in various circuit configura- 
tions. After considering several available active elements that are used 
for this type of switching, we shall modify the original argument to 
account for the frequency dependence of the nonideal active elements. 
A physical device differs markedly from an ideal one, and some of the 
latter results may even contradict the earlier conclusions. If the reader 
considers only the problem posed in each individual network, the argu- 
ments employed will be found to be consistent. 

11-1. Basic Circuit Considerations. The regenerative circuits treated 
in Chaps. 9 and 10 — bistable, monostable, and astable multivibrators — 
depended for their proper self-switching action on the charge stored in a 
capacitor (the single energy-storage element of the circuit) maintaining 
the tube or transistor cut off. Only after a sufficient amount of the 
stored energy had been dissipated did the active element reenter its 
active region; then, during recovery, the original stored energy was 
replenished through a positive source impedance. But in order to trans- 
fer the capacitor from a region of low potential energy (off) to one of high 
energy (recovery), the switch itself must contain an energy source. Thus 
a switching circuit might well be characterized as a device having two 
stable or quasi-stable states, i.e., the dissipative regions, and a single 
transition zone. 

Any two-terminal device with these attributes will, when combined 
with a capacitor or inductance, function as a multivibrator. Figure 11-1 
presents the idealized volt-ampere characteristic of one class of such 
devices. In regions I and III, the curve has a positive slope with a cor- 
responding positive incremental resistance. These are the dissipative 
segments. Region II is the negative-resistance portion of the character- 
istic, and it is here that the switching of circuit states takes place. 

Negative-resistance characteristics arise only through the use of active 
elements, which serve as the source of the available energy. Passive 

315 



316 



SWITCHING 



[Chap. 11 




Fig. 11-1. Idealized-negative-resistance 
characteristic (current-controlled) . 



elements, by their very nature, are always dissipative. Yet no physical 
device can be expected to have an unlimited range of negative resistance; 
to do so implies an infinite source of energy. Therefore each negative 
slope will always be flanked by two regions of positive resistance. 

Because the terminal voltage is a single-valued function of current, we 
classify the characteristic of Fig. 11-1 as a current-controlled nonlinear 

resistance (CNLR). On the other 
hand, any one of three values of cur- 
rent becomes possible upon exci- 
tation by an input voltage. The 
particular value that will flow is 
determined by the constraints im- 
posed by the external circuit, with 
this very multivaluedness permitting 
the several different circuit states. 
Since only the current is unre- 
stricted, in our switching circuit we must utilize an energy-storage ele- 
ment allowing instantaneous current changes during the switching process, 
i.e., a capacitor. 

An alternative form of the nonlinear resistance, the dual of the above, 
also exists. Its voltage-controlled characteristic is illustrated by the 
sketch of Fig. 11-2 (VNLR). To complete the duality, an inductance 
would be used instead of the controlling capacitor to permit rapid 
terminal voltage changes. 

Negative-resistance regions are exhibited by many devices. Current- 
controlled characteristics appear in point-contact transistors, in uni- 
junction and avalanche diodes, and 
in gas tubes. For examples of the 
voltage-controlled nonlinear resist- 
ance (VNLR), we may refer to the 
screen characteristics of a tetrode or 
a pentode, to the tunnel diode, and 
to the p-n-p-n transistor. In addi- 
tion, the use of feedback allows the 
production of almost any desired 
form of NLR. Generally voltage 
feedback furnishes the voltage-con- 
trolled characteristics and current feedback establishes those that are 
current-controlled. 

Behavior of the two classes of NLR are similar enough so that any 
conclusions derived from the examination of one class may be applied to 
the other. Of course, caution as to the duality relationship must be 
observed while interpreting the results. 




Fio. 11-2. Voltage-controlled nonlinear 
resistance. 



Sec. 11-2] negative-resistance switching circuits 



317 



11-2. Basic Switching Circuits. The basic form of the switching cir- 
cuit with which we shall be concerned is shown in Fig. 11-3. For the 
negative-resistance element, we will use the current-controlled character- 
istic previously given. In this circuit, the items of interest are the neces- 
sary conditions for stable operation at any one point and the mechanism 
of transition from one state to the other. We shall start the analysis by 
writing the single-node equation : 



tr = i n + ic = in + C 



dv n 



dt 



(11-1) 

and V nn - i r R = v n (11-2) 

Solving Eq. (11-2) for i, and substituting the result into Eq. (11-1) yields 

,dv„ 



Let 



(V nn - i n R) - », = RC 

v t =RC^ 
at 



dt 



(11-3) 
(11-4) 



However, F„ n — i„R is simply the static (d-c) load line which we can 
superimpose on the volt-ampere characteristic. The distance measured 



h 'n 






« 4 


t 


CNLR 


-r- *nn 










~° 









. 


"» 






B 








A i 








/c 




K.-A/ 


s"' 










'x 


^° 


V„„\ 


/ '" 


Eh 










^^Vnn-inR 

^^ Load line 



Fig. 11-3. Basic CNLR switching circuit. Fig. 11-4. Graphical solution of the cir- 
cuit of Fig. 11-3. 

from the known curve to the load line must be v ( , and as shown in Fig. 
11-4, it may be treated as a vector. 

The steady-state operating point, the intersection of the load line 
and the characteristic, is stable if, and only if, any momentary perturba- 
tion will create the conditions forcing a return to the original point. 
Since the volt-ampere plot represents the locus of all possible behavior of 
the NLR, any disturbance will shift the operating point along it, say 
from A to A'. Equation (11-4) defines the vector v ( , drawn from any 
point on the characteristics to the load line, as being proportional to the 
time rate of change of v n . At point A', this vector is negative, indicating 
that the terminal voltage across the capacitor must decrease with time. 
If instead the initial disturbance were from A to A", the vector would be 
positive, with the terminal voltage increasing with time toward point A. 



318 



SWITCHING 



[Chap. 11 




Only at A itself is v 4 zero, and only here do we have a stable operating 
point. 

Operation as a Monostable Switching Circuit. A voltage pulse, applied 
in series with the bias supply, momentarily shifts the load line to the 
position indicated by the dashed line of Fig. 11-5. While the pulse is 
present, the shifted load line establishes a new stable point at F. But v h 
as drawn from the original operating point A to the shifted load line, is 

positive, and therefore the terminal 
voltage increases toward point B. 
At the peak, vt is still positive, for- 
bidding travel down line segment 
BD. Operation must be restricted to 
the characteristics and must stabilize 
at point F. It can do so only if the 
operating point can transfer from 
segment EB to CD. The voltage 
across the capacitor may not change 
instantaneously, but the current can and does; it jumps from B to C. 
Along line segment CF, v< becomes negative. After the jump, the 
direction of the operating path is down toward point F. 

Removal of the input trigger permits the load line to return to its 
original position. Even after it does so, the terminal voltage, constrained 
by the capacitor to change slowly, remains on segment CD and travels 
toward point D. At D, v ( is still negative, preventing the traverse up the 
line segment DB. A second jump in current is indicated, this time from 
D to E. Following this jump, v ( becomes positive, and the operating 
path is finally back toward the original stable point A. 

In order to repeat the operating cycle, another trigger must be injected. 
The minimum-size pulse, for guaranteed triggering, is that value neces- 
sary to shift the load line so that it just clears the peak point B. 

As an alternative method of triggering, we can inject a current impulse, 
having an area of K amp-sec, into the common node of Fig. 11-3. KS(t) 
changes the voltage across the capacitor from the initial value of Va to 



Fig. 11-5. Monostable operation. 



»(0+) 



Va + — 



If y(0+) is greater than the peak voltage V B , the operating point will 
switch to the appropriate voltage coordinate along the extension of line 
segment CD. Recovery proceeds as before, from the new initial value 
established by the applied impulse. 

Since this system has one, and only one, stable condition, any disturb- 
ance, no matter how large, eventually results in the circuit returning to 
this point; it functions as a monostable device. 



Sec. 11-2] negative-resistance switching circuits 



319 



Bistable Operation. For operation as a bistable, the value of V„„ and 
the load resistance will have to be changed so that the circuit has the two 
stable points (A, G) indicated in Fig. 11-6. Upon first examination, H 
appears to be a third stable point, but once switching begins, the transi- 
tion will always be from A to G for positive triggers and from G to A when 
a negative trigger is applied. The operating path is in the direction of the 
arrows. Regardless of how or where the triggers are injected, it is 
impossible to establish a transition path to point H. 

A practical CNLR differs in important aspects from the ideal, and it is 
these very differences that preclude the establishment of point H as a 
third stable point (Sec. 11-8). In general, it is difficult to stabilize a 
system within its negative-resistance region; whenever there is an appre- 
ciable-size energy-storage element present, it becomes almost impossible 
to do so. 



B 


"" / 


y^^H 


JJ. / 


/ ° 

eA — • 


D 



Fig. 11-6. Bistable operation. 




Fio. 11-7. Astable operation. 



In effecting the change of state, the external trigger forces the operating 
point to move from its stable position to the peak of the curve, i.e., 
from A to B or from G to D. In sweeping through this distance, power 
proportional to the product of the changes in voltage and current will be 
dissipated, and this power must be supplied from the trigger source. 
We conclude that if the stable points are established closer to the peaks, 
the switching-power requirement imposed on the trigger generator will 
be much less severe. 

The sequence of the multivibrator's behavior in switching from one 
stable point to the other is identical with that discussed for the mono- 
stable circuit. 

Astable Operation. When the load line is so chosen that it intersects 
the volt-ampere characteristic only on the negative portion of the NLR, 
no stable point of operation exists. This case is illustrated in Fig. 11-7. 
A power balance, evaluated in the vicinity of the point of intersection J, 
would show that the energy supplied by the NLR is greater than that 
dissipated by the load resistor. The surplus energy, available for storage 



320 switching [Chap. 11 

by the capacitor, results in a voltage build-up toward either point B or 
point D. 

Once the peak is reached, the path of operation becomes BCDE, with 
the instantaneous current jumps taking place from B to C and D to E. 
We might observe that at all points on the characteristic, the value of v, 
is such as to direct the time rate of voltage change toward the two peaks. 

An alternative way of viewing the build-up phenomena is to recognize 
that in region II the negative resistance — R n and the load resistor R are 
in parallel. Surplus energy means that the parallel combination is still a 
negative resistance with the system's pole located at 

v = - C0F1F*3 (11-5) 

This pole lies on the real axis in the right half of the complex plane. 
Obviously, the circuit is unstable, having an increasing exponential for 
its transient response. Terminal voltage builds up until limiting occurs 
when the operating point enters the positive-resistance region. 

Instantaneous current jumps are forbidden in any physical CNLR 
circuit by the inductive component of the driving-point impedance. 
This modifies the path of operation, introducing the curvature shown 
by the dashed lines of Fig. 11-7. For very-low-frequency operation, the 
transition time is small enough so that the path approaches the ideal. 
But as the generated pulse duration decreases, the transition time 
becomes an appreciable portion of the total cycle and the locus constricts 
to the dashed curve sketched. At very high frequencies, the path may 
even turn into a small ellipse. 

Stability of Ideal CNLR Switching Circuits. For absolute stability, 
i.e., the system's pole remaining on the left-hand half of the real axis, the 
parallel combination of the ideal negative resistance and the external load 
resistor must be positive. 

_ R(-R n ) 
Kp ~ R + {~R n ) >U 

In the particular circuit of Fig. 11-3, where the graphical solution is 
given by Fig. 11-7, this condition can be satisfied only when 

R < \Rn\ (11-6) 

If the load is a short circuit, then regardless of the value of R n the parallel 
combination will always be positive. In addition, the now horizontal 
load line intersects the positive-resistance, as well as the negative- 
resistance, region of the CNLR and consequently establishes at least 
two stable operating points. Alternatively, an infinite-resistance load 



Sec. 11-3] negative-resistance switching circuits 



321 



could intersect only the negative-resistance portion, and it would cause 
astable behavior in the presence of any external capacity. It thus seems 
logical to refer to the ideal CNLR as a short-circuit stable device. Other 
considerations will cause us to modify this conclusion for a physical 
CNLR, and reference should be made to Sec. 11-8, where the topic of 
stability is treated in much greater detail. 

11-3. Calculation of Waveshapes. The calculation of the waveshapes 
of an ideal switching circuit (no parasitic energy-storage elements pres- 
ent) may be greatly simplified by 
representing each linear region of the 
NLR by a resistor in series with a 
battery. If we extend the straight 
lines of NLR until they reach the 
voltage axis, the point of intersection 
is the battery voltage and the slope 
of the line segment is the resistance 
value. Figure 1 1-8 shows the result- 
ant equivalent circuit for region I. 
Since the instantaneous current jump carries the operating point through 
the negative-resistance region, only regions I and III need be considered. 

Within each region, the exponential terminal response has, as its single 
boundary condition, voltage continuity across the current jumps of the 
circuit. The problem resolves itself into finding the boundary values and 
the time constants holding within regions I and III. 

At the instant that the operating point enters region I, the voltage 
across the capacitor is that of point E in Fig. 11-5, 11-6, or 11-7. It 
rises with a time constant r t toward the steady-state voltage of the 
TheVenin equivalent circuit across its terminals V\. 







1 I' \ 


-T V ' 


j_ 





Fig. 11-8. Piecewise-linear circuit used 
for the calculation of waveshapes. 



ti = C(ft [| ft,) 
Fi = BV t + R,V n 



fti + ft 



(11-7) 

(11-8) 



This voltage [Eq. (11-8)] is simply the value of the intersection of the load 
line with the extension of the line segment defining region I. In the 
astable circuit, Vi must always be larger than V B ; otherwise the output 
will never reach the voltage peak B. However, if the circuit is adjusted 
for bistable or monostable operation, and if it has a stable point along the 
segment EB, then 7 X will be the voltage at the stable point. Once this 
value is reached, sweep action ceases. 

At the peak, v n = V B and the operating point flips from B to C. 
Upon entering the new region (III), the cireuit of Fig. 11-8 changes; ft m 
and V IU replace fti and Vj. The terminal voltage immediately starts 



322 



SWITCHING 



[Chap. 11 



charging toward the new TheVenin equivalent voltage with a time con- 
stant T3. 

RVm + RmV n , 



V 3 = 



Riu + R 



t 3 = C(R || Bar) 



(11-9) 



Equation (11-9) defines the final value of any circuit having a stable 
point lying on line segment CD. For astable operation, V 3 < Vd- 
After reaching the peak, the operating path jumps to point E, and the 
cycle repeats. 



»E 



Id 



in 













1 
t 


1 
1 




7 A 

t * \ 
/ ' \ 


/ \ \ 
/ ' \ 

/ ! \ 




/ i \ 


r < \ 


/ %t 


r 
t 


i 
i 






"^ 


























^-^ 


«3 










t 


~ii>m 
















^^" 






«=--Tj- -| 















Fig. 11-9. Waveshapes produced in a CNLR switching circuit (astable operation). 

The current values at the switching points are simply those given by 
NLR characteristic at points B, C, D, and E. In order to find the steady- 
state values, we must consider the circuit when the capacitor is fully 
charged and the full current flows into the NLR terminals. In region I, 



and in region III, 



h 



/a = 



' nn 


- v 1 


Ri 


+ R 


r nn 


- Fin 



#in + R 



(11-10) 
(11-11) 



These values also correspond to the point of intersection of the extended 
volt-ampere characteristic and the load line. 

From the characteristic and from Eqs. (11-7) to (11-11), we have 
all the voltage and current values necessary to sketch the terminal 
waveshapes. Figure 11-9 shows the current and voltage waveshapes for 



Sec. 11^3] negative-besistance switching cibcuits 323 

an astable configuration, and Fig. 11-106 those appearing in the mono- 
stable circuit of Example 11-1. 

The duration of the two portions of the generated waveshape, evaluated 
by substituting the appropriate voltage values into the general time 
equation, is 

h = T-i In J'~ J/ (11-12) 

and U = t 3 In Z 3 ~ Y* (11-13) 

V z — V B 

In some NLR devices we may exercise a measure of control over the 
shape of the volt-ampere characteristic. Adjusting the ratio of Ri to Rm 
determines the ratio of pulse width to pulse spacing. Furthermore, if Ri 
is made very much larger than R m and if Fi is very much larger than V B , 
then the circuit becomes an almost linear sweep. Its fast recovery time 
is mainly dependent on Rm. 

We see that the greater the degree of control over the characteristic, 
the more versatile the system. Not all devices allow for wide latitude of 
adjustment, but even in those that do not, the characteristics might be 
modified by external diodes biased for conduction at the desired voltage 
levels (Chap. 2). 

Example 11-1. A CNLR having the volt-ampere characteristics given in Kg. 
ll-10a is used as a monostable pulse generator. Its external load consists of a l-/ti 
capacitor and a 2,000-ohm resistor in series with a 20-volt source, as shown in Fig. 11-3. 

Only two sets of coordinates are unknown: (1) the location of the intersection of 
the load line with the line segment defining region III, i.e., the single stable point A; 
(2) the intersection of the load line with the extension of segment I, that is, Vi and h. 
In order to find these points we first solve the graph for the three denning equations 

Region I: »„i = 110 + (5 K)i„ 

Region III: t,„ m - -12.5 + 250i„ 

Load line: v nL - 20 - (2 K)i„ 

By equating »»m to t>„£, we find that the stable point is located at 

1% = 14.45 ma and V, = —8.9 volts 

The other set of coordinates is 

1 1 12.85 ma and V, = 45.7 volts 

After the pulse is applied, the operating point jumps to point F, and i n and v„ start 
charging toward /, and V,. From Eq. (11-12), the time required to reach the peak is 

h = 1,430 X 10-«In ^ + ^ = 636 ^ec 
where t, - (2 K || 5 K)l ^f = 1,430 Msec. 



324 



SWITCHING 



[Chap. 11 



Upon jumping to point C, the circuit recovers toward the stable point A with 
the time constant 

t, = (2 K || 250)1 »i = 222 yusec 

Figure 11-106 shows the current and voltage waveshapes generated in this circuit. 
Note that the recovery portion of the wave is longer than the pulse duration and that 
the voltage rise is almost a linear sweep. 



(-20,10) K ->- 




&/tsec 



Fig. 11-10. Monostable operation, (a) CNLR characteristic and graphical construc- 
tion for Example 11-1; (6) current and voltage waveshapes produced after triggering. 

We might further note that the ratio of the recovery time to the 
pulse duration depends on the relative slopes of the two positive-resist- 
ance regions. For a fast pulse and a long recovery, the single stable 
point should be established in the high-resistance region. For a relatively 
fast recovery and a long sweep, the stable point must be placed in the 
low-resistance region, as it is in the above example. 



Sec. 11-4] negative-resistance switching circuits 



325 



11-4. Voltage-controlled NLR Switching Circuits. The basic switch- 
ing circuit associated with the VNLR is simply the dual of the one dis- 
cussed in Sec. 11-2. Figure 11-11 illustrates the resultant series switch- 
ing circuit together with the graphical construction holding when adjusted 
for monostable operation. 

By properly arranging the terms, we can write the input loop equation 
in a form that will present the steady-state solution as well as indicate 
the permissible time variation. 



(V--i.«)-».=L^-v. 



(11-14) 



It follows from Eq. (11-14) that the intersection of the load line (V nn — 
i„R) with the characteristics «„ is the only possible stable operating point 





(a) 
Fig. 11-11. (a) Basic VNLR switching circuit; (6) monostable path of operation. 

(di„/dl = 0). Any perturbation from this point creates a time rate of 
change of current that forces the operating point back to the original 
stable point (point A in Fig. 11-116). Thus the path of operation, 
location of the stable points, calculation of minimum trigger amplitude, 
and the calculation of circuit waveshapes are similar to those discussed 
for the CNLR. The major difference is that now the current is con- 
strained to change slowly, but this only supports the duality relation- 
ship, i.e., the interchange of the circuit's voltage and current response. 

The second important difference in the ideal VNLR concerns the 
condition which must be satisfied for absolute system stability. Here 
the external load appears in series with any internal incremental resist- 
ance. In order to prevent the system pole from migrating into the right 
half plane, the total series combination must remain positive. 



or 



R. = R + (-«,) > 

R > Rn 



(11-15) 



This stability condition holds only for the ideal VNLR circuit. A 
practical device differs in important respects from the ideal, and as with 
the CNLR, the stability criterion will be reexamined in Sec. 11-8, 



326 



SWITCHING 



[Chap. 11 



11-5. NLR Characteristics — Collector-to-base-coupled Monostable 
Multivibrator. As an example of how all single-energy-storage-element 
switching circuits may be treated from the viewpoint of their NLR 
characteristics, we shall now reexamine the collector-base-coupled mono- 
stable multivibrator of Chap. 9. Since the logical place to look for the 
negative resistance is across the energy-storage-element terminals, Fig. 
ll-12a shows the basic circuit with the timing capacitor omitted. In 
our analysis we shall make use of the equivalent circuit drawn in Fig. 
11-12&. Both transistors are shown in their active region, and as each 
is driven into either cutoff or saturation, the model must be accordingly 
modified. 



+25 v 




(a) 




lO* 3 '"- 



R 
•vVSA- 



-WV- 



\ZVV I I 



(b) 
Flo. 11-12. (a) Monostable multivibrator; (6) equivalent circuit. The timing 
capacitor is normally connected across the driving-point terminals. 

The circuit is initially adjusted so that in its normal state T 2 is fully 
on and T\ is cut off. Even though the storage time of the saturated 
transistor limits the switching time, this configuration is considered 
because of its inherent simplicity. Further simplification arises from the 
following conditions: 

1. /S is very large; therefore /S + 1 = 0. 

2. R » R 2 , and R 3 » R 2 . 

Generally, three or four line segments are sufficient to define the com- 
plete voltage-ampere characteristic. In finding these segments attention 
should be focused on the controlling current, i a . By starting from a high 
positive value of input current and considering the permissible change of 
state as the current decreases to, and below, zero, the possible operating 
regions are: 



Sec. 11-5] negative-resistance switching circuits 327 

Region 1 Tt saturated and 2\ off 

Region 2 Both transistors in their active region 

Region 3 Tt active and 2\ saturated 

Region 4 Tt off and T\ active 

• Region 5 T s off and Ty saturated 

At most, two of the last three regions can exist. Once T\ saturates, 
the decreasing input current may still drive Ti through its active to its 
cutoff region. Thus there is the possibility of operating over regions 3 
and 5. On the other hand, if T 2 is driven into cutoff while Ti is still 
active, the final bounding region will be number 4. 

In each region, we shall solve for the equation of e„ as a function of i„, 
in the form 

e„ = E { + R,4a (11-16) 

Ei is the voltage intercept found at i a = 0, just as if the defined region 
remained invariant. Ri is the incremental driving-point resistance in the 
region under examination. A sequence of such straight lines presents, 
in a piecewise manner, the complete volt-ampere characteristic. In order 
to find the boundaries of the individual segments, we solve for the coordi- 
nates of the intersection of the appropriate two lines. We might also con- 
sider the physical limitations imposed by the active elements employed 
in establishing these bounds. 

Region 1 — T 2 Saturated and Ti Off. Here 4i = and e C 2 = 0, and 
hence 

e a = -Eu, + i„R! (11-17) 

The lower limit of this region, where both Ti and T 2 become active, 
occurs once 

e C 2 = E» - PhtRi = (11-18) 

where «« = ^ + i a (11-19) 

From Eqs. (11-18) and (11-19) we find that this zone is limited to 

• ~» ^ bb _ El* 
%a > $Ri R 

And since R < /3i?2, region 1 holds down to some negative value of i a . 

Region 2 — 2*i and T 2 Active. From the solution of the circuit's loop 
equation, when in is given by Eq. (11-18) and ibi = e e %/R%, 

Note that once a transmission path involving the gain of both transistors 
exists, the slope of the volt-ampere characteristic becomes negative. 



328 switching [Chap. 11 

Here the positive feedback provides the required energy source. This 
negative-resistance region is bounded when Tj turns off at iu = 0. And 
the corresponding constraints on t« are 



e» _ E& > «■ > _ Ea 

jiRz R R 



(11-21) 



Region 3 — Ti Active and Ti Saturated. The conducting collector-base 
diode of T\ and the emitter-base diode of Ti now short-circuit the input 
terminals. The input equation becomes 







/ 



(11-22) 



Region 4 — Ti Off and Ti Active. Under the conditions holding in this 
region, 



ibi = 



R% 



Ri 



and therefore e a = fiE» ^ + i a (R + Ri) 

Tx finally becomes saturated for 



(11-23) 



\Rt R\J 



Region 5 — Ti Off and Ti Saturated. Ti being an open circuit and Ti 
a short circuit, the defining equation becomes 



e a = Eu, + i*R 



(11-24) 



Example 11-2. In this problem we shall calculate and plot the characteristic for 
the typical transistor monostable multivibrator of Fig. ll-12a. 

As the first step, the defining equation for each region is evaluated : 



Region 1. 
Region 2. 
Region 3. 
Region 4. 
Region 5. 



e. = -25 + (1 K)t« 
e„ ^ -36.6 - (81 K)i. 
e„ = 

e. = 29.2 + (51 K)i„ 
e„ = 25 + (50 K)i a 



Eq. (11-17) 
Eq. (11-20) 
Eq. (11-22) 
Eq. (11-23) 
Eq. (11-24) 



By plotting the five straight lines, as in Fig. 11-13, we can determine the permissible 
circuit states. As the current decreases, we note that the segments traversed are 
1, 2, 3, and 5. There is no possible way to reach line segment 4, and it therefore repre- 
sents a forbidden mode of operation for this particular circuit. 

One of the two remaining coordinates of interest is the location of the valley point. 
To find the intersection of lines 1 and 2 we equate (11-17) to (11-22), with the result 



T 36.5 - 25 

' " (81 + 1) K 
V, - -25.14 volts 



-0.14 ma 



Sec. 11-6] negative-resistance switching circuits 320 

The current intercept of region 5 is found from Eq. (11-24) by setting ft, ~ 0, 

7 " = oi! = - a5ma 

All the resistors have been included in the base amplifier, and consequently the 
voltage axis is also the steady-state load line. The circuit remains at its single stable 
point (i« ■= 0, «. = 25 volts) until a trigger forces it into its unstable state. Transition 
is directly from region 1 to region 4. Here the slope is simply the timing resistance 













t 


.Qp\ 


wits 




















































i i 
i i 
/ i 


























i 
i 


i 


























// 


























i 


i 
/ 
/ 


























i y 


•3 
















*a.t 


na 


-1 


0-0 


8-o'y\ 


-0.4- 


0.2 





2 


4 


6 


8 1 



















-10 
—15 
• 20 


















4, 
/ 


7* 




\2 




















/ I 


w 






















/ 


h 






ivK 




-25 








1 
















T 
\ 
\ 
\ 


-30 






~ Sl( 


)pe 1 


0001 


1 














\ 
\ 

\ 













































FlO. 11-13. CNLR characteristics of the collector-base-coupled monostable multi- 
vibrator of Example 11-2. 

(50 K), and it is here that the output pulse is generated. When the circuit switches 
back to segment 1, it enters the low-resistance (1-K) recovery portion of the cycle. 
Since the load line is the voltage axis, the time constants are the product of the 
slope of the line segment and the timing capacitor. Moreover, the steady-state 
charging voltage is the voltage intercept; the steady-state current must be zero. 
All the information necessary to calculate the voltage across, and current through, C 
is contained in the volt-ampere characteristic of Fig. 11-13. 

11-6. Some Devices Possessing Current-controlled NLR Character- 
istics. The Point-contact Transistor. Illustrating, as it does, the com- 
plete range of problems which must be treated in applying a practical 



330 



SWITCHING 



[Chap. 11 



negative-resistance device to switching circuits, the point-contact tran- 
sistor is the first device exhibiting CNLR characteristics with which we 
are concerned. This was not only the first semiconductor device in 
which a negative-resistance region was observed but also the first device 
which was extensively used in two-terminal switching circuits. 

Positive current feedback from the collector back to the base, further 
emphasized by padding the base with a large series resistor, produces 
a negative driving-point impedance at either the emitter or collector 
terminals, with the most useful characteristic appearing at the emitter. 
The transistor circuit and its large-signal equivalent model are shown 
in Fig. 11-14. 

First the low-frequency, static driving-point characteristic may be 
evaluated by examining the three possible regions of transistor operation, 



<ti t 




Ecc^: 



Fig. 11-14. (a) Point-contact transistor connected to develop a CNLR driving-point 
impedance; (b) large-signal T model. 

cutoff, active, and saturated. Under cutoff conditions (i e < 0) the cir- 
cuit model would be modified by removing the current generator, by 
replacing the emitter diode with its reverse resistance r re , and by replacing 
the collector diode by its large reverse resistance r„. From the reduced 
equivalent circuit, we find that the linear equation defining this region is 



v, = ie[r r . + (r b + flx) || (r« + jR 2 )] - 



r b + Ri 



n + r„ + Ri + Rt 



S i.r r , 



2?cc 

(11-25) 



where r„ :» (r 6 + R x ) || (r« + Ri) and where the polarity of E cc is taken 
into account (Fig. 11-14). The large value of r rc appearing in the 
denominator usually makes the value of V p close to, but slightly less than, 
zero. 

In the active region, the emitter diode is forward-biased, and we might 
conveniently replace it by a short circuit. Also, r rc >J> (Ri + #2 + rt), 
allowing the representation of D c by an open circuit for the purpose of 
computing the input impedance. As a consequence, the driving-point 



Sec. 11-6] negative-kesistance switching circuits 331 

equation becomes 

v, £* i.{\ - a){n + Ri) - V p (11-26) 

where V p is the voltage intercept at t, = 0. Within this region the 
regeneration that develops the negative driving-point impedance is 
supplied by the greater-than-unity value of the emitter-to-collector 
current generator (a > 1). 

The final zone, complete transistor saturation, is characterized by 
heavy conduction through both diodes shown in Fig. 11-146. For the 
defining equation we need only consider Ri, r b , R it and Ea,. 

v. = [(r 4 + flx) || RJi. - ■ ri +z 1 + Rl E « C 11 " 27 ) 

To find the coordinates of the valley point, i.e., the point separating 
the active from the saturation region, we simply solve Eqs. (11-26) 
and (11-27) simultaneously. But before beginning the calculation, even 
further simplification is possible. We are primarily interested in the 
negative-resistance portion of the characteristic, and in order to ensure 
an appreciable negative-resistance value, Ri !2> n [Eq. (11-26)]. If 
we also neglect the small value of V p in Eq. (11-26), the location of the 
valley point will be given by 

T ~ ~ E " V ~ ~ Rl ^ ~ °i) Ecc m^S"! 

U = fli(l - a) - aR 2 K "-J?i(l - a) - «R 2 K ' 

One obvious design difficulty is that the valley-point coordinates are a 
function of the transistor a and vary rather widely between individual 
transistors. We can compensate by making Ri adjustable. 

Figure ll-15a shows the three regions of circuit behavior together 
with their particular resistance values. In a practical transistor, the 
nonlinear characteristics would introduce some curvature at both peaks. 

The readily accessible additional elements of the device used to gen- 
erate the CNLR characteristic permit the extraction of an output 
waveshape at a point well removed from the pulse timing network of 
R and C (Fig. 11-3). If the output is taken at the collector, with the 
circuit so adjusted that the critical timing occurs during the cutoff 
interval, then any loading of the collector will have only an extremely 
small second-order effect on the pulse duration. 

Within the active region the proportionality between collector and 
emitter current is simply a. Outside this region, the transistor is either 
cut off, with the collector current dropping to its small 7 c0 value, or 
saturated, with the incremental current relationship now becoming 

ic^-s^pi. (11-29) 

til T til 



332 switching [Chap. 11 

To this we would add the current contributions from the bias batteries, 
which of course depend on the particular circuit configuration. But since 
we already know the location of the valley point of i, [Eq. (11-28)], 
the corresponding collector current is simply 

I„ = al v 

For current flow above this value, the slope of the transfer characteristic 
is given by Eq. (1 1-29). The information we now have yields the current 
transfer relationship shown in Fig. 11-156. 




«c 


, 













iu ie 




y» 






*«r 


















(b) 


3% 



Fig. 11-15. (a) Point-contact-transistor CNLR driving-point characteristic and (6) 
emitter-collector current transfer characteristic. 

Since the transistor switches through the active region very rapidly, 
when we evaluate the collector output waveshapes (both current and 
voltage), only the cutoff and saturation regions need be considered. 
This calculation will be left to the reader in his solution of individual 
problems. Furthermore, the additional elements permit the injection 
of the input trigger at the transistor base. By taking advantage of the 
amplification afforded within the switching transistor, the trigger require- 



Sec. 11-6] negative-resistance switching circuits 



333 



merits become much less severe. And under cutoff conditions the trigger 
source is also isolated from the timing circuit. 

The p-n-p-n Junction Transistor. The p-n-p-n junction transistor also 
exhibits CNLR characteristics (Fig. 11-16), and like the point-contact 
transistor, it depends for its negative-resistance region on the positive 
current feedback from the output back to the base. In fact, within the 



"i 




Fig. 11-16. p-n-p-n circuit and its CNLR characteristic. 

point-contact transistor a p-n junction is established between the collector 
contact and the base, thus leading to a p-n-p-n structure. This similarity 
allows the immediate application of all results derived for the point- 
contact transistor to the new device. 

We might alternatively represent the p-n-p-n transistor by a coupled 
configuration of two junction transistors, a p-n-p and an n-p-n (Fig. 
11-17). When both are within their active regions, the collector cur- 
rent of T\ is injected into the base of T%, where it is amplified by the 



11 «JU * J 




J -1$ 



"1 




(!-«)»,- pah \ l fl i 



Fig. 11-17. The two-transistor equivalent configuration of the p-n-p-n circuit. Junc- 
tion currents are indicated. 

large base-collector amplification factor 0. Finally, the collector of T t , 
directly coupled to the base circuit of T\, supplies the positive current 
feedback necessary to develop the negative input impedance. The 
approximate current flow in each branch of Fig. 11-17 is indicated in the 
interest of clarity. Note that the total composite base current is nega- 
tive. By substituting the value of 0, we find this current to be 



in = in — id = [! — (« + P<*)]ii = (1 - 0)ti 



(11-30) 



334 



SWITCHING 



(Chap. 11 



Except that the larger amplification factor replaces a, the current 
given by Eq. (11-30) is of the identical form as the current that would flow 
in the base circuit of the point-contact transistor (Fig. 11-146). Thus 
the input impedance of this circuit, when it is in its active region, may 
be expressed as 

B ta S (1 - fi)Ri (11-31) 

The cutoff frequency of this transistor can be made quite high, allowing 
satisfactory operation in switching circuits generating l-jusec or smaller 
pulses. 

The Unijunction Transistor. A silicon unijunction transistor consists 
of a bar of n material with a p emitter junction located somewhat above 
the center point of the bar (at the 60 to 75 per cent mark). Under the 
emitter cutoff condition, the conductive current flow through the high- 
resistivity silicon establishes a voltage drop from the emitter to base 1 




Fig. 11-18. Unijunction transistor circuit and input volt-ampere characteristic. 
proportional to the spacing between these points. The emitter remains 
back-biased until the input driving voltage exceeds the conductive drop. 
Once the emitter begins conduction, the minority carriers injected into 
the n bar travel to base 1 and increase the charge density in this region; 
they effectively decrease the driving-voltage drop required for a given 
current flow. Consequently, a negative input resistance appears between 
the emitter and base 1 (Fig. 11-18). 

An equivalent circuit for the unijunction transistor which explains, at 
least to a first approximation, the behavior of this device in its various 
operating regions is presented in Fig. 11-19. While it remains cut off, 
the effective input impedance is simply the reverse emitter diode resist- 
ance, about 500 K. But once the emitter voltage reaches the back-bias 
voltage yEu,, where 



r»i 



and 



V, = q£» 



the diode conducts. The defining equation now becomes 
v. = nEtb + [r. + (1 - y)r hl || r b2 ]i. 



(11-32) 



Sec. 11-6] negative-besistance switching circuits 



335 



If, in addition, account were taken of the small voltage drop (0.7 volt) 
appearing across the emitter junction, then the peak point of Fig. 11-18 
would be shifted slightly up and to the right. 

The current amplification factor y varies with frequency in much 
the same manner as does a in the point-contact transistor. But y further 
depends on the mobility of the 
majority and minority carriers, and 
therefore it will change with the 
current flow, from a value of ap- 
proximately 3 at low current density 
to unity at saturation. As a result, 
the negative-resistance region is ex- 
tremely nonlinear. In the model 
drawn in Fig. 11-19, all regions are 
linearized by assuming a constant y. 

The firing of the diode D 2 charac- 
terizes the saturation region, and the 

full generator current yi, flowing through it serves to maintain full 
conduction. Replacing D 2 by a short circuit permits the writing of the 
appropriate saturation-region equation: 

v.= V.+ (R, || r 61 || r„)i. (11-33) 

R. II r*i 




Fig. 11-19. Model describing the be- 
havior of the unijunction transistor. 



where 



V, 



R. II r hl + n, 2 



Eu 



A large measure of control over the slopes and intercepts of the indi- 
vidual regions is afforded by the insertion of additional resistance in series 
with bi and 6 2 . Any resistance we add to 6i has the most pronounced 
effect on the saturation region, and that included at 62 mainly increases 
the magnitude of the negative input resistance. 

Currently available unijunction transistors are limited to low-speed 
switching applications by their y cutoff frequency of only 0.7 or 0.8 mega- 
cycle. The correspondingly large input inductive component of the 
input impedance (Sec. 11-8) will adversely affect the switching path even 
at low frequencies. Rather than the expected large instantaneous cur- 
rent jumps from the peak and valley, the path of operation becomes 
almost elliptic. 

Avalanche-region Operation. Among the other devices exhibiting 
current-controlled negative-resistance regions are the space-charge 
diode and the avalanche-breakdown transistor. Their current multipli- 
cation depends on the high reverse voltage establishing an extremely 
high electric field intensity. This field dislodges valence electrons, and 
by creating additional electron-hole pairs, it increases the current density. 
Avalanche-region operation is widely used for very fast switching circuits 



336 



SWITCHING 



[Chap. 11 



since there is no inherent minority storage time. However, the negative- 
resistance region is quite narrow and the circuit will generate only a small- 
amplitude signal. 

11-7. Some Devices Possessing Voltage-controlled Nonlinear Charac- 
teristics. The tetrode, one of the earliest devices found to exhibit volt- 
age-controlled negative resistance, was the predecessor of many of the 
secondary-emission multiplier tubes in current use. Because the energy 
supplied to the external energy-storage elements often resulted in an 
undesirable oscillation, operation within this region of the plate volt- 
ampere characteristic was usually avoided. Realizing now its usefulness 
for switching and oscillator applications, and in order to round out our 
discussion of NLR devices, we shall discuss the tetrode along with several 
other voltage-controlled devices. Electrical-engineering literature refers 




Fig. 11-20. Tetrode VNLR plate characteristic. 

to the particular tetrodes used for their negative-resistance character- 
istics as dynatrons. 

At very low values of plate voltage, the total cathode current flows 
to the screen, which is the tube element at the highest potential. As its 
voltage is raised, the plate receives an ever-increasing percentage of the 
total current flow. The electrons attracted to the plate strike it with 
more force, because of additional energy acquired as they are accelerated 
through the higher potential present. Eventually, their energy on 
impact becomes sufficient to cause emission of the valence electrons of 
the plate material. Since the screen is maintained at a constant poten- 
tial, somewhat higher than that now appearing from plate to cathode, the 
secondary-emission electrons will travel from the plate back to the 
screen, causing the current at the plate to decrease, even though its 
voltage continues to increase (Fig. 11-20). Also present is the possibility 
that the plate current may become negative as a consequence of the large 
emission from the plate surface. 

As the plate voltage continues rising, up to and even beyond the 
screen voltage, the secondary electrons begin returning to the plate. 



Sec. 11-7] negative-resistance switching ciecuits 



337 



Under these circumstances, the screen current falls as the rising current 
at the plate bounds the negative-resistance zone with the expected 
dissipative region. 

Pentode VNLR Characteristic — The Transitron. A pentode may be 
so connected that it also will display a VNLR driving-point character- 
istic, but at its screen rather than at its plate (Fig. 11-21). At low 
values of screen voltage, the suppressor, highly negative with respect 
to the cathode, completely cuts off the plate current. The tube func- 
tions as a triode, and the total cathode current, which now flows to the 
screen, obeys Child's %-power law. In this region, the screen resistance 
is identically that of the tube connected as a triode, r c2t . As the screen 
voltage continues rising, the suppressor voltage E c2 - E 3 which must rise 
with it eventually reaches a point where it allows plate conduction. Any 




Fio. 11-21. Pentode circuit for VNLR transitron operation and resultant screen-circuit 
volt-ampere characteristic. 

further rise in the screen voltage increases the total tube current, and 
simultaneously, through its effect on the suppressor, E e2 increases the 
percentage of the current flowing to the plate. Less current flows in 
the screen circuit, and the volt-ampere characteristic now exhibits a 
negative-resistance region. 

Once the suppressor becomes positive and draws current, it saturates. 
The current flow through R 3 maintains the suppressor voltage at only a 
few volts positive with respect to the cathode. Above this point, the 
ratio of plate to screen current remains reasonably constant, and both 
now increase with the rising screen voltage. Since the conditions are 
identical with those treated for the phantastron in Sec. 6-9, the screen 
resistance in this region will be 

r C ip = r cJl (l + p ) 

where r eit = resistance in first region considered 
P = ratio of plate to screen current 
If we were now to turn back to the phantastron, which was discussed in 
Sees. 6-8 and 6-9, we would see that the switching action occurring in 



338 



SWITCHING 



[Chap. 11 



the screen circuit depends on its negative-resistance characteristic. Since 
the screen configuration is essentially of the form shown in Fig. 11-21, 
the normal load line should be selected for bistable operation. The two 
circuit states are (1) when the suppressor cuts off the plate (region I 
of Fig. 11-21) and (2) when the suppressor saturates (region III of 
Fig. 11-21). The normal position of the phantastron is in region I, 
requiring a positive pulse for switching. However, the circuit itself sup- 
plies the negative pulse necessary to switch the screen back from region III 
to region I. The presence of the varying control-grid signal means that 
the path of screen operation will be along the composite of several of the 
family of characteristic curves, instead of along a single curve. 



1 
2.0 


. 


























V 


3 

= .1.0 


«-7 




M> 






c 


7 


CO/ 




Y 



























5 


1( 


)0 1! 


)0 2( 


30 2 


JO 3( 


X) ' 




m 

(a) 
Fig. 11-22. (a) Tunnel-diode volt-ampere characteristic; (b) model holding in the 
negative-resistance region — L, is due to lead inductance, R, represents the lead resist- 
ance and ohmic losses in the semiconductor, and C is the junction capacity. 

Tunnel Diode. The tunnel diode, which was discovered by Dr. L. 
Esaki in 1957, consists of an extremely thin p-n junction formed between 
two heavily doped regions (large amounts of added impurities) of a semi- 
conductor. The shape of the volt-ampere characteristic (Fig. ll-22a), 
which exhibits a voltage-controlled negative-resistance region for small 
forward biases, can be explained from a consideration of the electron- 
wave propagation through the junction boundary. Because of the 
heavy doping there exist relatively large numbers of conduction electrons 
in the n material and a wide range of empty states in the p material. The 
electron wave propagates freely within each region but cannot tunnel 
through the potential barrier at the junction unless the energy level on one 
side is matched by an equivalent empty state on the other side. During 
the transition the electron wave is attenuated, while the energy is con- 
served. Thus the junction must be thin in order for the electron wave 
to have an appreciable probability of transmission. 



Sec. 11-7] negative-eesistance switching circuits 339 

When the diode is back-biased, the energy level of the electrons in the 
n material is lowered below that of the free electrons in the p material. 
The reverse current, which is completely due to tunneling, can increase 
without limit. In this region, the volt-ampere characteristic looks 
exactly like that of a conducting diode. 

An applied positive bias increases the potential energy of the electrons 
in the n material. As a consequence the forward current will continue to 
increase until the complete range of free states in the p material is 
matched by the tunneling electron waves. Any further increase in the 
forward bias raises the energy level of the free electrons above that of the 
empty states and is, therefore, accompanied by a decrease in the terminal 
current. This is the negative-resistance zone. Eventually the increas- 
ing bias causes the injection of the minority carriers, the diode conducts 
in the normal manner, and the current again increases with increasing 
voltage. 

The major advantage of the tunnel diode over all other negative- 
resistance devices is the high speed of the current transmission. The 
velocity of propagation approaches that of light. There is, however, a 
large capacity (20 to 60 n/ii) associated with the junction as well as series 
resistance and lead inductance shown in Fig. 11-22&. These parasitic 
elements tend to slow the switching time. In spite of this, the large 
value of negative conductance permits the generation of pulses having 
rise times of less than 10 -10 sec. Sinusoidal oscillation at frequencies in 
excess of 4,000 megacycles is also possible (see Chap. 15). 

It should be noted that the negative-resistance region is quite narrow; 
the peak is normally located between 50 and 100 mv and the valley point 
at 150 to 500 mv. Thus, if the junction is designed for large peak cur- 
rents, the negative-resistance value will be extremely low. For example, 
approximately —2 ohms is measured in the active region of a gallium 
arsenide tunnel diode having a peak current of 100 ma. When we exam- 
ine a diode whose maximum current is less than 1 ma, we observe that the 
average negative resistance has increased to a few hundred ohms. 

Since the impedance of the power supply is of the same order of 
magnitude as the negative resistance, it is extremely difficult to bias 
properly and to stabilize the very-low-resistance devices in their negative- 
resistance region. Even a small amount of lead inductance may lead to 
undesired switching or spurious oscillations. 

A two-terminal tunnel diode does not afford the same flexibility in 
obtaining an isolated output as do the three-terminal active devices. 
Consequently care must be taken that the external load will not disturb 
the operation of the timing or switching circuits. In order to minimize 
possible interaction when multiple stages involving tunnel diodes are to 
be interconnected, some type of unilateral decoupling must be used. We 



340 switching [Chap. 11 

may even be forced to associate a junction transistor with each tunnel- 
diode circuit. Because the actual rise is limited by the slowest stage, 
many of the advantages of the tunnel diode are negated. In some cir- 
cumstances it becomes practical to employ fast-responding diodes to 
decouple the individual switching circuits. Diodes whose conduction 
is due to tunneling, but which do not exhibit negative resistance, are ideal 
for this purpose. 

Point-contact and jMi-p-n Junction Transistor. A VNLR driving- 
point impedance will also be developed at the base of the point-contact 

transistor and at the base of its junc- 
° c tion equivalent, the p-n-p-n transis- 

tor. Instead of merely discussing 



-W- 



-,t 



4^ 






O - — *~~| this device qualitatively, we shall 
JJ L define the complete circuit behavior 

by following our usual procedure of 
examining the operation with respect 
Ecc~=~ to the increasing independent vari- 
able, in this case Vi. When the drive 
voltage is highly negative, the tran- 
Fig. 11-23. Model for VNLK driving- s i s t r will be completely saturated. 
£or. imPedanCe ° f point " contact tran " Since this condition is characterized 

by heavy conduction through both 
of the diodes in the model of Fig. 11-23, the input node equation, which 
is written by superposition, becomes 

H = -EJBi + EJ3* + v l (G l + G 2 ) (11-34) 

where Gi = l/i?i and G 2 = 1/Ri. 

As »i rises and as the transistor enters its active region, the emitter 
current decreases with increasing driving voltage. 

i. = (E M - »i)Gi 

The input current contribution from E cc that normally flows through 
r rc will be small enough so that it can be ignored, thus allowing the 
following defining equation to be written for the active region: 

ii (1 - a)i. = -(1 - a)Gi(E m - »i) (11-35) 

From Eq. (11-35) we conclude that a > 1 is the only required criterion 
for a negative driving-point impedance at low frequencies. 

When the p-n-p-n transistor is used in place of the point-contact 
transistor, (3 would replace a in Eq. (11-35). 

Once the high positive value of Vi cuts off the transistor (vi > E„), 
the circuit enters into its third region. With r„ > R\ and r« > R%, the 



Sec. 11-8] negative-resistance switching circuits 341 

input equation is of the same form as Eq. (11-34); only the conductance 
terms must be changed to conform to the new conditions. 

*i = -E ee g„ + E cc g rl . + vxig,. + g rc ) (11-36) 

A plot of the three characteristic equations appears in Fig. 11-24. 




Fio. 11-24. Driving-point volt-ampere characteristic of point-contact transistor— base 
input. 

11-8. Frequency Dependence of the Devices Exhibiting NLR Charac- 
teristics. Up to this point we have discussed only the static behavior of 
the various devices that exhibit negative-resistance regions. But to 
utilize these elements properly, their frequency limitations must also be 
known. Some of these were treated briefly in earlier chapters, e.g., the 
rise and fall times and the minority storage time of the solid-state devices. 
The major item which we must yet consider is the effect of frequency on 
the transistor parameters. 

In each case, the negative input resistance developed in the active 
region depended on the current amplification of the active element 
employed. Equations (11-26), (11-31), and (11-32) defining the nega- 
tive-resistance regions are all basically alike. Of course, the controlled- 
current-source parameter is a in one case and or y in the other two, but 
this is simply a detail dependent on the particular device considered. In 
all three, the term of interest is of the form 

Rin = (1 - a)Bi 

where a is now used as a general current amplification factor. 

The parameter most affected by frequency is this very amplification 
factor. Equation (11-37) expresses its approximate variation, and we 
note that a decreases at a rate of 6 db per octave as w rises above the 
3-db point u c . 



a(a>) = 



«o 



(11-37) 



1 + jw/o>c 

The a cutoff frequency /„ usually ranges from 100 kc to 2 megacycles 
in the unijunction and the p-n-p-n transistors, up to about 100 mega- 



342 



SWITCHING 



[Chap. 11 



cycles in point-contact and junction transistors that aie specifically 
designed for high-frequency operation. 

If we substitute Eq. (11-37) into the general negative-impedance term, 
the resultant expression becomes 



5 in S (l 1 + 7o,/ w J 



Bi 



/ u c 2 a \ , . 

= Ri I 1 ,-; — ; ) + .?«- 



\ c R\cto 
2 + to 2 



(11-38) 



The first part of Eq. (11-38) represents the resistive portion of the input 
impedance, which will remain negative up to 

&h = W e V a O — 1 

Above this frequency, the various transistors can no longer sustain their 
negative-resistance characteristics and are unable to supply any energy 
to the external load. It follows that switching operations would have 
to be restricted to the range of frequencies below <a h . 

The second term contributes an inductive component to the driving- 
point impedance. First of all, this prevents any instantaneous current 
jumps. Secondly, it may also resonate with the external capacity, 

generating an almost sinusoidal 
waveshape, provided that the cir- 
cuit is biased within the active 
region. 

Figure 11-25 shows an equivalent 
driving-point network which repre- 
sents the incremental behavior of 
the device within its active region 
and which, of course, also satisfies 
Eq. (11-38). At low frequencies, 
the input inductance is effectively 
Fig. 11-25. Equivalent driving-point net- a short circuit, leaving only the 
work of a CNLR in the active region and nega ti ve -resistance term. At ele- 
the external load. yated frequencies the pre sence of 

the additional energy-storage term complicates the network and forces us 
into a reevaluation of the conditions to be satisfied for absolute circuit 

stability. 

Stability of the Nonideal CNLR Circuits. The problem of stability can 
be neither considered nor defined without treating the complete circuit 
before us. Instability simply means that sufficient surplus energy exists 
in the system to produce an increasing exponential response in any 
energy-storage element present. But if the external network also con- 
tains dissipative elements, then their effect on the system may even 







Sec. 11-8] negative-resistance switching circuits 343 

change one having an energy surplus into one that is completely dissi- 
pative. The particular conditions necessary to ensure this happening 
are those that produce absolute stability and prevent any regenerative 
switching. 

Consider, for example, the network (Fig. 11-25) that represents the 
input characteristic of the current-controlled devices of Sec. 11-6. The 
external load consists of a parallel combination of R and C together 
with a bias battery whose function is to set the intersection of the load 
line and the NLR characteristic within the proper region. In the follow- 
ing discussion we shall assume intersection of the negative resistance. 
Since the NLR is current-controlled, for stability the poles of the current 
function must not be allowed to he in the right half plane. And this 
corresponds to restricting the zeros of the impedance function [Eq 
(11-39)] to the left half plane. 

RtRCLp' + [(«! + R)L + CRRfaoil - a„)]p 

= + «oRi[R + g x (l - «„)] 

(pL + aoRJipRC + 1) t 11-39 } 

In order to guarantee this restriction, all the coefficients of the poly- 
nomial of the numerator of Eq. (11-39) should have the same sign. This 
leads to the two inequalities Eqs. (11-40) and (11-41), which we must 
satisfy to ensure absolute circuit stability. 



R > -Ri(l - oo) (11-40) 

Ri + R 
RRi(ao — l)w, 



C < Efl^-V (H-41) 



The value of L given in the model of Fig. 11-25 was substituted while 
solving for the condition expressed in Eq. (11-41). 
^ When both inequalities hold, the system cannot operate as a switching 
circuit. Once C is larger than the minimum value given by Eq. (11-41), 
and if Eq. (11-40) is still satisfied, the circuit becomes free-running. 
At the bounding value of C, the zeros of the impedance function he 
on the imaginary axis; for C less than this value, the zeros are in the left 
half plane and the system is stable. As C increases, the roots move 
into the right half plane toward the real axis (Fig. 11-26). If the zeros 
are complex conjugates lying close to the imaginary axis, the time 
response of the astable system is sinusoidal. This particular mode of 
operation will be discussed in Chaps. 14 and 15. 

For bistable operation R must be less than the magnitude of the nega- 
tive resistance; i.e., Eq. (11-40) must remain unsatisfied. The load line 



344 



SWITCHING 



[Chap. 11 



•will now intersect the positive-resistance regions of the volt-ampere 
characteristics. Both zeros of the impedance function will always lie 
on the real axis, one in the right half plane and the other along the nega- 
tive real axis. As C increases, both zeros move to the right, the negative 
root approaching zero as a limit and the positive root approaching 

Ph = Wc(«0 — 1) 

for large C. Thus the switching speed, through the negative-resistance 
region, is limited primarily by the frequency response of the active 
element employed. 





-,. 






r-l 


).7i 


■I m axis 

II 
















- 1 


P 


1 
















^ 


1 1 1 
^1,500/^f- 












































































1.5 


























































































> 


,-0.4 






















































































• ( 


).3 






























































































).2 






























































































).l 
















































10,000 
///jf 










4,500 












' 10,000 


































-0.3 -0.2 -0.1 







"0.1 


*(*2 


0.3) 


0.4 1 0T5 


06' 


07 n 
































xin e 








R e axis 












1 














































1 
-0.2 

l 


























































































1 
-0.3 

l 
























fl!=lK 

«=2 

■u c = 10 6 

i?=2K 




















































1 
-0.4 

I 










































































1 
-0.5 
































































































































r--H 






































If 




1 
M).7 





































Fig. 11-26. Path of the zeros of Z(p) [Eq. (11-39)] as a function of C. Specific circuit 
values are given above for astable operation. 

The stability condition presented in Eq. (11-40) is exactly opposite 
to the condition found when we discussed the ideal NLR in the earlier 
portion of this chapter [Eq. (11-6)]. This does not really contradict 
the earlier discussion, since a short circuit or a low resistance across 
the input terminals of the equivalent network of Fig. 11-25 results, 
essentially, in the inductively controlled behavior of the VNLR (Fig. 
11-11). Furthermore, when &>„-»<» or when L = 0, that is, when the 
circuit approaches the ideal, the stability conditions found from Eq. 
(11-39) will reduce to the single equation given in Eq. (11-6). 



Sec. 11-8] negative-resistance switching circuits 



345 



We conclude that a physical negative-resistance device cannot support 
a stable point within the negative-resistance region unless the two 
inequalities of Eqs. (1 1-40) and (11-41) are satisfied. Point H of Fig. 1 1-6 
(the load-line intersection with the negative resistance) will, in general, be 
unstable. Away from the vicinity of this point, the system rapidly 
stabilizes at one of its two inter- 
sections in the completely dissipative 
regions. 

Stability conditions are seen to 
depend on the particular circuit con- 
figuration. In order to avoid ambi- 
guity, it is much better to refer 
always to the device used in terms 
of its single-valuedness, i.e., current 
or voltage, rather than its conditions 
for stability. Furthermore, when 

any changes are made in the external circuit, the reader must re-solve 
for the conditions necessary to ensure absolute stability. 

VNLR Stability. In Sec. 11-7 we saw that the basic form of the nega- 
tive conductance developed in the active region was 

Y in = (1 - a)G, 

This equation is of the identical form with the negative-impedance func- 
tion derived for the CNLR devices. It follows, from Eq. (11-38), that 
the substitution of a(<o) into the admittance term results in 




Fig. 11-27. Equivalent input network of 
the VNLR holding during the active 
region and the external load. 






Jc 2 + O) 2 



(11-42) 



Equation (11-42) yields the equivalent input network given in Fig. 11-27, 
which we note to be simply the dual of the network of Fig. 11-25. 

To ensure that the poles of the controlling voltage function will not 
he in the right half plane, the zeros of the admittance function must 
be restricted to the left half plane. The two required conditions for 
absolute system stability become 



G 

L < 



> -G x (l - a ) 
G x + G 

GGi(a - l)<o. 



(11-43) 
(11-44) 



Equations (11-43) and (11-44) are the duals of Eqs. (11-40) and (11-41) 
previously derived. And as before, when neither equation is satisfied, 
the load line also intersects the positive-resistance regions, thus establish- 
ing bistable circuit operation. Satisfaction of only Eq. (11-43) allows 
astable behavior for L larger than the minimum value of Eq. (11-44). 



346 



SWITCHING 



[Chap. 11 



Just as with the CNLR, this circuit will also exhibit an almost sinusoidal 
oscillation for small values of L. 

11-9. Improvement in Switching Time through the Use of a Nonlinear 
Load. Generally, if fast signals are to be generated, and if narrow pulses 
are used for triggering, operation within the saturation region of semi- 
conductor devices must be avoided. The long time delay introduced 
by the minority-carrier storage precludes all but the slowest switching 
intervals. In order to accomplish this restriction, we rely on nonlinear 
load lines which are developed with the aid of appropriately biased diodes. 

With one region forbidden us, both of the remaining circuit states 
must be used in a bistable; one stable point will be situated in the cut- 
off zone, and the other will lie in the negative-resistance region. A simple 
bistable circuit for high-speed switching, featuring this arrangement, is 
shown in Fig. 11-28. It makes use of a single external diode to insert 





Fig. 11-28. Stabilization within the negative-resistance region — bistable operation. 

the necessary break in the load line, ensuring that it intersects the volt- 
ampere characteristic only in these two regions. 

The absolute stability of the point established in the completely 
dissipative cutoff region, by the diode's conduction (point A), is not 
open to question. But for a true bistable, we must justify the stability 
of point F, which lies on the negative-resistance portion of the char- 
acteristic. At this point the large load resistance R definitely satisfies 
one of the two stability conditions [Eq. (11-40)]. If the stray capacity 
is kept low enough, or if the parameters of the CNLR satisfy the second 
condition [Eq. (11-41)], or an equivalent condition if the device used is 
neither a point-contact nor a p-n-p-n transistor, then point F will also be 
absolutely stable. From Eq. (11-41) we see that the smaller the value 
of negative resistance, the larger the capacity that can be tolerated with- 
out changing the stable to an unstable point. 

Switching in the circuit of Fig. 11-28 will always be from A to F and 
from F back to A. We depend on the combination of the stray capacity 
and the inductive component present in the driving-point impedance 
of the CNLR to slow the system response enough so that the path of 



Sec. 11-9] negative-resistance switching circuits 



347 



operation will never enter the saturation region (see the dashed path of 
Fig. 11-28). 

A stable-operation Stabilization. An alternative technique for prevent- 
ing the transistor from being driven into saturation, illustrated in Fig. 
11-29, is most widely applied in astable systems. This circuit configura- 
tion requires the insertion of a resistor fi 2 in series with C to stabilize 
the intersection occurring in the negative-resistance region. Suppose 
that the charge stored in C back-biases the diode while the operating 
point is at A. Then the additional resistance increases the net circuit 
dissipation to a point where it will exceed the net energy supply. 

In order to find the conditions which must be satisfied to stabilize 
point A when the diode is back-biased, we must solve the complete 
model of the system. Representing the CNLR by the model of Fig. 1 1-25, 







il 








1 


. 




i?> 






C5 


v m -. 


i 


n 


CNLR 


+ ** 












D -i 








Fig. 11-29. Stabilization within the negative-resistance region — astable operation. 

the two conditions which will ensure absolute circuit stability in the 
negative-resistance region are 



R > -Ri(l - «„) 
p > (oco ~ l)RR\ 



Ri + R 



Ri(l - a ) + R [Ri(l - ao) + R]Cu c 

where the negative resistance is R\{1 — <*o)- The first condition is the 
same as expressed in Eq. (11-40). From the second condition we see 
that the smallest value of iE 2 that will maintain the circuit stable, regard- 
less of the value of C, is a resistance equal in magnitude to the parallel 
combination of R and (1 — ad)Ri. This makes the net resistance across 
the energy-storage element positive. 

However, C eventually charges, and at the instant that the voltage at 
its lower terminal reaches zero, the diode again conducts. The circuit 
becomes astable, and the circuit flips clockwise to point B. As discussed 
in Sec. 11-3, the capacitor charges toward the Th6venin equivalent 
voltage across its terminals, eventually reaching the peak, V p . 



348 



SWITCHING 



[Chap. 11 



In this circuit, the voltage across the terminals of the CNLR can 
change instantaneously because this change will not appear across C 
but will be coupled by C to R 2 . Any voltage drop, no matter how small, 
back-biases the diode. The operating point thus jumps from the peak 
to the now stable point A, and the diode is driven negative by this 
same voltage change. C again charges, with its bottom terminal rising 
from —(Vp — Va) toward V and with the approximate time constant 

r 2 s (R» + R || - R n )C 

It eventually reaches zero, the diode conducts, the circuit again becomes 
astable, and the cycle repeats. The approximate voltage waveshapes 



V A 









/* 


/ l 






[ 




( 


1 j i 


t 


t 
V D 

(V„-V A ) 


__ 


/* 


[/ 



Fig. 11-30. Voltage waveshapes appearing at the input of CNLR and across the 
diode in the circuit of Fig. 11-30. 

generated across the diode and at the input to the CNLR are sketched 
in Fig. 11-30. 

An arrangement similar to that shown in Fig. 11-29 might also be 
used in a monostable circuit. In this case, returning Ri to a small nega- 
tive voltage — V, instead of to a large positive one, will keep the diode 
back-biased and guarantee the stability of point A. After an external 
trigger forces the diode into conduction, the circuit momentarily becomes 
free-running and switches clockwise into the cutoff zone. If the circuit 
parameters are properly adjusted, the capacitor charging current will 
maintain diode conduction during the complete interval that the operat- 
ing point remains in this region. 

Subsequent to reaching the peak value of the CNLR, any drop in 
voltage, coupled by C, will turn off the diode and reestablish point A 
as the single stable point. The operating point drops from the peak 
to point A, and the circuit recovers. Since the steady-state voltage 
across the diode is — Vj the diode remains back-biased and the operating 
point finally stabilizes at A . 



SEC. 11-10] NEGATIVE-BESISTANCS! SWITCHING CIBCOTTS 



349 



11-10. Negative-impedance Converters. The basic approach taken 
in developing negative-input characteristics, through the use of active 
elements, is illustrated by the two typical block diagrams of Fig. 11-31. 

In the circuit of Fig. 11 -3 la, when we write the input node equation, 
assuming an ideal voltage amplifier, 

ei — Avei 
we readily arrive at the recognized form of the Miller input impedance 



Zi. = 



(11-45) 



If the voltage amplification within the active region is greater than 
unity, then the input impedance becomes negative. Under the particular 



»1 




A» 


S— 








t 

ei 

1 




f V 


**c > 










" 




1 



(a) (b) 

Fia. 11-31. Basic negative-impedance converters (NIC), (a) Voltage-controlled cir- 
cuit; (6) current-controlled configuration. 

conditions where A r is set equal to +2, we obtain the following very 
convenient result: 

Z* = -Z L (11-46) 

In general, Ay is a two-stage amplifier having definite frequency char- 
acteristics and the over-all response will not be in as simple a form as 
Eq. (11-46). 

The simplest possible operation of the basic current-controlled circuit 
of Fig. 11-316 is where the complete input driving current flows through 
the short-circuited input of the ideal current amplifier. Under these 
circumstances the input loop equation is 

ei = (*i - i»)Z L = fi(l - A C )Z L (11-47) 

and by making A e = 2, 

Ziu = J- = — Zi 

However, if only a small percentage of the input current flows directly 
to the output, the load current becomes 



t'i = Ki-L — A, 



■H 



350 



SWITCHING 



[Chap. 11 



And if this term is substituted into Eq. (11-47), the input impedance may 
be written 



Z„ = (K - A C )Z L 



(11-48) 



where K < 1 and A c is the forward current gain. 

For example, in the p-n-p-n transistor, A c = 0. Consequently, when 
these terms are substituted into Eq. (11-47), the resultant expression is 
identical with that given in Eq. (11-31): 

Z„ = (1 - 0)Z L 

All the devices that were discussed in Sees. 11-6 and 11-7 could just 
as easily have been analyzed by converting their equivalent circuits 
into the appropriate block diagrams. By doing so we would have lost 
the insight that was gained in examining the actual device. The major 
role served by the block diagrams is to indicate the general conditions 

toward which we must design if 
(l-oti)/! we wish to develop negative-input 

characteristics. 

Consider, for example, the stand- 
ard negative -impedance -converter 
(NIC) circuit of Fig. 11-32. Except 
for the additional bias resistors and 
batteries, it is of the identical con- 
figuration of the composite transistor 
circuit used to describe the behavior 
of the p-n-p-n transistor. Once we 
recognize that the output load is 
Rl II R, we can also identify this cir- 
cuit as a practical form of the basic 
impedance converter of Fig. 11-316. 
Only a small fraction of the input current flows directly to the output, 
and we must therefore evaluate both K and A c in Eq. (11-48). The 
current through the direct transmission path is simply (1 — ai)i' lt and 
since c*i is very close to unity, this term becomes insignificant compared 
with i' x . K in Eq. (11-48) may be taken as zero. Thus the input 
p-n-p stage acts as a simple current amplifier having a gain, from the 
emitter to the collector, of approximately unity. 

The output current of T x divides between the collector load R and the 
input impedance to T 2 of R(Pz + 1): 




Fio. 11-32. A practical negative-imped- 
ance converter — current-controlled oper- 
ation. 



^2 = 



t'xfl 



03, + l)R + R 



NEGATIVE-RESISTANCE SWITCHING CIRCUITS 

and the current gain to the emitter of Ti becomes 



A B = 



^1 



351 



(11-49) 



ft + 2' 

where the approximation holds for a high-gain transistor. 

Substituting K = and A c = 1 into Eq. (11-48), we find the effective 
input impedance to be 

Z[ n = -(R\\Rl) 

But this is paralleled by the input resistance R, which cancels the nega- 
tive term due to — R. The only remaining term is that dependent on the 
load Rl\ 

Z in = -Rl (11-50) 

This same result can also be obtained by noting that the current flow 
through Rl \\ R is — i[. Since a short circuit exists through the emitter 
and base of T lf the voltage drop across the load must be identically 

»i = ~i'i(RL II R) 

By taking into account the role of the input resistor R, the input imped- 
ance will be as given by Eq. (11-50). 

Of course, here also the pure negative input resistance would exist 
only at very low frequencies. At the higher frequencies, the frequency 
dependence of a t and /3 2 will introduce the additional inductive component 
shown in Fig. 11-25. 



PROBLEMS 

11-1. A certain CNLR with the characteristics given in Fig. 11-33 is used in the 
basic sweep of Fig. 11-3. The load resistance R is 4 K, and the external timing capac- 
ity is 2 ni. Plot the current and voltage waveshapes after the circuit is triggered if 
V n n is as given below. In each case state the area of the current impulse which must 
be injected into the node to cause switching. 

(O) Vnn = 40 volts. 

(b) V„„ = -40 volts. 
11-2. (a) What limits of load resistance 

will permit bistable operation of the 
CNLR of Fig. 11-33? 

(6) The load resistance used with CNLR 
of Fig. 11-33 is 10 K. What ranges of 
V„„ permit operation as a monostable 
circuit ? 

(c) Repeat part o if the load resistance 
is now 2 K. 

11-3. A 5-jif capacitor is connected directly across the input terminals of the CNLR, 
which has the characteristics of Fig. 11-33. Sketch the current and voltage wave- 
shapes, labeling them with respect to time constants, voltage and current values, and 
times. 




Fig. 11-33 



352 



SWITCHING 



[Chap. 11 



11-4. At t = the network N of Fig. 11-34 is activated by the voltage impulse 
indicated. Sketch i n and v» as functions of time, labeling clearly all break points and 
time constants. 

11-6. (a) Give the range of external resistance that will make the circuit of Fig. 
11-34 astable when the battery voltage is between the limits of 4 < F„„ < 10 volts. 

(6) Sketch the output waveshapes, labeling them completely, when B = 10 ohms 
and V„ n = 10 volts. 



2S(t) 




Fig. 11-34 

11-6. Calculate and plot the CNLR characteristics seen when looking across C in 
the monostable circuit of Fig. 9-38 (Prob. 9-14). 

11-7. Plot the CNLR characteristics for the circuit whose values are given in Fig. 
11-12 when R = 100 K and flj = 25 K. All other parameters remain as before. 
Superimpose this plot on a copy of Fig. 11-13. 

11-8. (a) Evaluate and plot the transfer characteristic, i.e., collector voltage of T t 
versus i a , for the monostable multivibrator (Fig. 11-12) discussed in Example 11-2. 

(6) Sketch one complete cycle of i a after the circuit is triggered. Using the curve 
found in part o, draw to scale 1 cycle of e C 2. 

11-9. Plot the volt-ampere characteristic for the multivibrator of Fig. 11-12 when 
R> is increased to 100 K. Compare your results with Fig. 11-13. Which regions are 
most affected by the increase in i? 3 ? What does it do to the modes of operation? 

11-10. Calculate and plot the volt-ampere characteristic seen across the inductance 
in the circuit of Fig. 11-35. Using this plot, sketch and label the inductive current ii 
after a trigger is applied. 



+20 v 




/3-60 <100 



11-35 



11-11. The point-contact transistor of Fig. 11-14 has a = 1.2, n = 200 ohms, and 
r„ = 150 K. Assume that the peak point of the input negative-resistance character- 
istic is located at the origin. Specify all circuit parameters necessary to set the valley 



NEGATIVE-RESISTANCE SWITCHING CIRCUITS 



353 



point at —10 volts and 10 ma. Sketch the driving and transfer characteristics, 
labeling all slopes. 

11-12. Plot the magnitude and phase of the input impedance of the point-contact 
transistor in its active region as a function of a. iJi = 5 K, « = 1.5, and u c = 10*. 

11-13. Assume that the p-n-p-n transistor of Fig. 11-17 is composed of two indi- 
vidual units, each having a = 0.98. In all other respects these transistors are ideal 
units. Plot the input volt-ampere and the ««2 versus ii characteristics of this device. 
Specify the break points when Ri = Ri = 1 K and E = 10 volts. Make all reason- 
able approximations in your calculations. 

11-14. The unijunction transistor shown in Fig. ll-36o may be represented 
approximately by the volt-ampere characteristic of Fig. 11-366. Sketch the wave- 
forms of V. and /, as functions of time, labeling clearly all break points and time 
constants. 



l/rf=T= 




■=LE kh 




via ni 

Fig. 11-36 



11-15. The unijunction diode of Fig. 11-18 has the following parameters: rn = 
140, r« = 60, r, = 30, y = 2.5, and R, = 10 ohms. The back-biased input imped- 
ance is 50 K. 

(a) Plot the input volt-ampere characteristics when Ebb = 20 volts. 

(6) Repeat part a when a 2,000-ohm resistor is inserted in series with 6 S and a 
1,000-ohm resistance in series with 6i. 

11-16. The point-contact transistor of Prob. 11-11 is used in the configuration of 
Fig. 11-23. R } = ijj = 5 K, and the reverse resistance of both the collector and base 
is 500 K. 

(a) Find E cc = E„ necessary to produce a current swing of 2 ma when the circuit 
operates as an astable device. 

(6) The external conductance is twice the limiting value. What is the smallest 
battery in series with the inductance that will allow the circuit to free-run? What 
is the largest series battery? v 

(c) If the series inductance is 100 mh, what is the smallest cutoff frequency of the 
transistor for satisfactory operation? Where are the poles and zeros of Y(p) located 
under this condition? 

11-17. (a) Prove the equivalency of the driving-point network of Fig. 11-25. 

(6) Derive Eq. (11-39) and verify Eqs. (11-40) and (11-41). 

(c) Show that as L —> 0, the stability condition is satisfied by Eq. (11-6). 

11-18. (a) The tunnel diode of Fig. ll-37a is connected to an external circuit 
consisting of a 10-mh inductance, a 1-ohm load resistance, and a 250-mv bias source. 
Approximate the characteristic by three line segments, and plot the terminal-voltage 
waveshape. 

(b) If R, = 0.3 ohm, L, = 2 pii, and C, = 50 wit in the active-region model of Fig. 
11-22, what is the highest frequency of operation? 



354 



SWITCHING 



[Chap. 11 



(c) Express the absolute stability conditions for the general tunnel-diode model of 
Fig. 11-226. 

11-19. A tunnel diode having the characteristic of Fig. ll-37a is used as a coinci- 
dence gate. The circuit appears in Fig. ll-37b. Positive input pulses applied at e% 
and ea have an amplitude of 1.5 volts and a duration of 0.1 iiaec. Plot the output 
voltage under the following conditions: 

(o) Eu, = 0, «i or e% present. 

(6) En, = 0, «i and ei simultaneously applied. 

(c) Ebb = 250 mv, ei or e 2 present. 

(d) Ebb = 250 mv, ei or ei simultaneously applied. 



, 


k 


















30 










| 












20 

> 

10 









































































— >• 




J 61 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

v d , volts 



(b) 



(a) 



Fig. 11-37 



11-20. Figure ll-38a shows a five-segmented approximation to a tunnel diode's 
volt-ampere characteristic. This particular device is employed in the switching cir- 




KS(t) 



10 mh 



:5oo 



t 



(b) 



Fig. 11-38 



NEGATIVE-RESISTANCE SWITCHING CIRCUITS 



355 



cuit of Fig. 11-386. A voltage impulse having an area K of 40 X 10~* volt-sec is used 
for triggering. Sketch and label the output voltage for both positive and negative 
impulses under the following conditions: 

(a) JS» = 0. 

(fc) En - 1 volt. 

11-21. A train of 1-volt positive and negative pulses, 0.1 /isec wide and spaced 
2 /isec apart, is applied to the circuit of Fig. 11-39. D\ and D 2 are tunnel diodes whose 
volt-ampere characteristic is linearized in the manner shown. Da is a fast-acting 
decoupling diode and may be considered ideal for the purposes of this problem. 

(o) Sketch and label the output voltage over a 10-/isec interval. 

(6) Repeat part a when D» is removed and the coupling is directly through the 
100-ohm resistor. 

200 mv 
100/ih; 



?lv 



•IK 



Input 

pulse o — VvV- 

train 300 



-Wv- 
100 



200 < 



-M- 



3CA, 




Fio. 11-39 

11-22. The terminal characteristics of a certain CNLR are shown in Fig. 11-40. 
Also shown is the equivalent circuit for the device in its active region. 





NLR (static) 



Active region 
Fig. 11-40 



External circuit 



(a) When the external network is connected by closing switches at t = 0, deter- 
mine in as a function of time. (Hint: Solve for the poles of the network in its active 
region and also for the steady-state component of i„; then write i n = itrtmient + »'.» and 
evaluate the necessary arbitrary constant.) 

(b) How does the circuit function if C is reduced by a factor of 2? If it is increased 
by a factor of 10? Give a qualitative answer and describe the expected terminal 
waveshape. 

11-23. (a) Repeat Prob. 11-14 when the 1-^if timing element is shunted by a Zener 
diode which conducts for V. > 8 volts (Fig. 11-28). 

(6) To what value must C be reduced before point F becomes the second stable 
point? 



356 switching [Chap. 11 

11-24. Prove that the insertion of a resistor in series with C (as in Fig. 11-29) can 
make a normally unstable circuit stable. [Hint: Return to the driving-point net- 
work of Fig. 11-25 and find the conditions necessary to restrict the zeros of Z(p) to 
the left-half plane when iJj is in series with C. Compare these with the stability 
conditions given by Eqs. (11-40) and (11-41). Equation (11-40) is satisfied and 
(11-41) is not when R t is removed.] 

11-26. In the negative-impedance converter of Fig. 11-32, Bl ■= B = 10 K, 
Vi = Vs = 10 volts, and |3i = /Ss = 50. Calculate the input volt-ampere character- 
istic. Specify the slopes and intercepts of each line segment. Find the current 
transfer characteristics from the input to the load, Rl. 



BIBLIOGRAPHY 

Anderson, A. E.: Transistors in Switching Circuits, Proc. I BE, vol. 40, no. 11, pp. 
1541-1548, 1952. 

Beale, I. E. A., W. L. Stephenson, and E. Wolfendale: A Study of High Speed Ava- 
lanche Transistors, Proc. IEE {London), pt. B, vol. 104, pp. 394-402, July, 1957. 

Ebers, J. J.: Four-terminal p-n-p-n Transistors, Proc. I BE, vol. 40, no. 11, pp. 1361- 
1365, 1952. 

Esaki, L.: Letter to the Editor, Phys. Rev., vol. 109, pp. 603-604, Jan. 15, 1958. 

Farley, B. G.: Dynamics of Transistor Negative Resistance Circuits, Proc. IBE, 
vol. 40, no. 11, 1497-1508, 1952. 

Hall, R. N.: Tunnel Diodes, IBE Trans, on Electron Devices, vol. ED-7, pp. 1-9, 
January, 1960. 

Leak, I. A., and V. P. Mathis: The Double-base Diode: A New Semi-conducting 
Device, IRE Conv. Record, pt. 6, p. 2, 1953. 

Linvill, J. G. : Transistor Negative Impedance Converters, Proc. IRE, vol. 41, no. 6, 
pp. 725-729, 1953. 

Lo, A. W. : Transistor Trigger Circuits, Proc. IRE, vol. 40, no. 11, pp. 1531-1541, 1952. 

Merrill, J. L.: Theory of the Negative Impedance Converter, Bell System Tech. J., 
vol. 30, no. 1, pp. 88-109, 1951. 

Shea, R. F.: "Transistor Circuit Engineering," John Wiley & Sons, Inc., New York, 
1957. 

Shockley, W., and J. F. Gibbons: Introduction to the Four-layer Diode, Semiconduc- 
tor Prods., January-February, 1958, pp. 9-13. 

Suran, J. J., and E. Keonjian: A Semiconductor Diode Multivibrator, Proc. IRE, 
vol. 43, no. 7, pp. 814-820, 1955. 



CHAPTER 12 

THE BLOCKING OSCILLATOR 



Most of the multivibrators discussed in Chaps. 9, 10, and 11 depended 
for their timing on the energy-storage element maintaining the tube or 
transistor off for the required interval. Because an active element is 
essentially an open circuit within its cutoff region, the output pulse is 
available only at a relatively high impedance level and with a low power 
content. Furthermore, stray capacity slows the fast rising and falling 
edges, preventing the generation of extremely narrow pulses. One 
method of overcoming these disadvantages is to time the pulse within 
the high-current low-impedance saturation region. 

The blocking oscillator is one circuit which is so designed. Only 
a single active element is necessary, with a specially designed transformer 
used for timing as well as for phasing the regeneration. Because this 
deceptively simple circuit is so widely used, in computers, radar, tele- 
vision, etc., it will be treated in some detail. In doing so, we extend the 
approximation methods of analysis to multiple-energy-storage-element 
systems. We shall also be obliged to consider the problems posed when 
some of the passive components exhibit nonlinearity. No attempt will 
be made to obtain an exact solution, but the analysis will adequately 
explain the role of the various elements and the functioning of the circuit. 

12-1. Some Introductory Remarks. Figure 12-1 shows one possible 
form of the blocking oscillator, a collector-emitter-coupled circuit, 
together with its general piecewise-linear representation. As possible 
alternative configurations, the transformer may couple either the collector 
or the emitter to the base. In brief, the sequence of monostable opera- 
tion is the following. The transistor is initially back-biased, and in 
response to an externally applied trigger, it is forced into the active 
region. Provided that the transformer has the proper turns ratio and 
connections (introducing a sign change), the net loop gain will be posi- 
tive and greater than unity. Regeneration drives the circuit into 
saturation. The transistor acts as a switch, which operates with a 
minimum amount of input energy and which connects Em, across the 
primary of the transformer. 

The full duration of the saturation interval is controlled, as is all 

357 



358 



SWITCHING 



[Chap. 12 



timing, by the energy-storage elements present (in this circuit by the 
magnetizing inductance L m and the coupling capacitor C). When gener- 
ating long pulses, an evaluation of the width may be further complicated 
if the large current build-up in the magnetizing inductance L m drives 
the transformer into saturation. For short pulses, the timing will be 
influenced by the rate of flux penetration into the core (Sees. 12-3, 13-2, 
and 13-3). The solution of the complete problem is quite complicated, 
involving as it does multiple modes of energy storage, core properties, 
and hysteresis. 




Ideal transformer 



R» r u , R L » r n 

Fig. 12-1. Collector-emitter-coupled blocking oscillator and complete piecewise-linear 
model. 



Eventually the circuit reenters the regenerative region and switches 
back off. The oscillator recovers and remains off until the next trigger 
is applied. 

The blocking-oscillator transformer is a very important component, 
so critical, in fact, that the circuit is designed about its characteristics. 
By interleaving the coil windings and by using a high-permeability core 
material, the leakage inductance and stray capacity are minimized, 
and consequently so is the rise time. For example, a transformer 
designed to generate a 1-itsec pulse may have a nominal magnetizing 
inductance of 1.0 mh compared with a leakage inductance of only 20 to 
50 ^h. As it is quite small physically, the stray capacity would only be 
10 wi . 

The transformer usually includes one or more tertiary windings which 
are used to couple the generated pulse to the isolated external circuit. 



Sec. 12-2] the blocking oscillator 359 

12-2. An Inductively Timed Blocking Oscillator. The following sim- 
plifications are now possible: 

1. The input impedance of the transistor in the active and saturation 
regions is so very low that C, will have a negligible effect on the rise time. 

2. The coupling capacitor C is very large, and we may therefore assume 
that its terminal voltage remains constant over the entire pulse interval. 

3. Both R and Rl are much greater than rn- 

4. The transformer core never saturates. 

Using the above assumptions, the incremental model holding during 
the active region reduces to the one shown in Fig. 12-2o. The two node 
equations are 

(7 c + i) e °-i ei = -°*« (12 - lo) 

- -L Cc + (~ + -i~) «i = (12-16) 

Substituting i. = — ei/nru, the solution of these equations yields the 
system's pole, which is located at 

p^_ r,(l-an) (12-2) 

From Eq. (12-2), we conclude that the condition which must be satisfied 
if the circuit is to be regenerative (a pole located in the right half plane) is 

an > 1 

and that this corresponds to a loop current gain greater than unity. 

Before the circuit enters into its active region, the value of the collector 
voltage is Em,; immediately after the pulse turns the transistor on, the 
collector drops to zero. The full supply voltage appears across L e , and 
the transistor saturates rapidly at an extremely low current. To find 
the time required, we must know the starting point of the exponential 
build-up. But this depends on the energy content of the excitation 
pulse, and since, in general, it is unknown, the problem is not completely 
defined. For a rough order of magnitude, we can approximate the 
initial switching time by 2.2 time constants of the positive exponential of 
Eq. (12-2). 

f i S 2.2 7 ^-- (12-3) 

{an — l)r c 

With a good pulse transformer Eq. (12-3) may even yield a time of less 
than 10 -8 sec. The turn-on time of the transistor may be very much 
longer, and if it is, it will predominate — the solution of Eq. (12-3) will 
be totally without meaning. 

Once the transistor saturates, the models of Fig. 12-2& and c will be 
used to define the collector and emitter current build-up. Because 



360 switching [Chap. 12 

L. «C L m , the actual double-energy problem will be treated by the meth- 
ods of Chap. 1, i.e., by assuming that the current build-up in L, is com- 
plete before that of L m starts. The waveshapes obtained from this 
approximation are sketched in Fig. 12-4. 




Fig. 12-2. Models holding for the various regions of blocking-oscillator operation — 
timing due to the transformer's magnetizing inductance. Waveshapes appear in 
Fig. 12-4. (a) Active-region incremental model; (6) models for the initial portion of 
the saturation region; (c) models for the final portion of the saturation region; (d) 
model for the recovery region. 

The primary voltage exponentially approaches a peak value of — En, 
with a time constant due to the leakage inductance and the input resist- 
ance of the transistor reflected through the ideal transformer. 



T2 



L« 



where ru is the saturation value of the input resistance. 



THE BLOCKING OSCILLATOR 



361 



Sec. 12-2] 

Since the voltage across C remains constant at E a , the net drop across 

rn will reach a peak of 

Eg 

n 



Eem — E a 



and the corresponding value of emitter current is given by 

E, 



= ±(^-E a ) (12-4) 

r n \n ) 

For saturation I m > 0, and therefore the circuit must be designed so that 



em ~~ 



E ^>E a 
n 



If this condition is not satisfied, the transistor will immediately switch 
back into its active region and no pulse will be generated. 



v 



«/,, 



hi ■: 



C/< >hm 



le***ei 



0' -. 



i«-0 



Fig. 12-3. Transistor collector volt-ampere characteristics — grounded-base connection 
— showing the path of operation of the blocking oscillator. The transition time 
between the various points is indicated. 

The primary current, which flows into the collector, is related to the 
emitter current by the transformer turns ratio n. 



Iim — — — - 



(12-5) 



This segment of the path of operation lies between points B and C in 
Fig. 12-3. The time required for the rise; from 10 to 90 per cent of the 
final value, as defined in Eq. (1-32), is 



h = 2.2r j = 2.2 - 



(12-6) 



Since ru is quite small, in order to minimize the rise time, L, must also be 
small. 

Next the magnetizing current starts building up. If the small resist- 
ances of the conducting collector diode and the transformer primary 



362 switching [Chap. 12 

are neglected, the model reduces to the one shown in Fig. 12-2c. The 
constant voltage across L m produces a linearly increasing magnetizing 



current 



£>« (12-7) 



which increases the total collector current, driving the operating point from 
C toward D in Fig. 12-3. When i c finally reaches al em , the net collector 
diode current drops to zero and the transistor reenters its active region. 
Since the emitter current remains almost constant over the range of 
interest, from Eqs. (12-4), (12-5), and (12-7), we obtain 

ic = Ji. + im = — (— - E^\ + ^5 t (12 -8) 

run \n ) L m v ' 

By setting i c equal to ale„, we find that the final portion of the pulse 
lasts for 



h-L, na ~ 



«_=ll(l_*KA (12 . 9) 

n 2 ru \ E»J v ' 



Because h must be real for any combination of terms, Eq. (12-9) again 
proves that the conditions which must be satisfied for the proper circuit 
operation are na > 1 and nE a < E bb . In this mode, the segment 
defined by the linear current build-up in the magnetizing inductance 
constitutes the major portion of the desired pulse (that is, W^>ti) and 
Eq. (12-9) essentially defines the complete pulse width. 

The transistor switches back off in much the same manner as it turned 
on, and the circuit begins its recovery. As the current in L„ decays, the 
change in the direction of di m /dt produces a large backswing of voltage 
which, coupled by the transformer, aids in back-biasing the emitter. 

From the model of Fig. 12-2d, we see that the primary voltage would 
be given by the product of the peak value of the magnetizing current and 
the reflected resistance. 

E mm = I mm 7i>R = (ocI em - I lm )n*R (12-10) 

The collector voltage jumps from zero to 

E cm = Etb -f- E mm 

with the emitter voltage going to 

Em 



mm 



E em = E„ + 

n 

Recovery toward the initial conditions is with the fast time constant 

'• - k < 12 - n) 



Sec. 12-2] the blocking oscillator 363 

Instead of the current and voltage decaying monotonically toward 
the steady-state values, the magnetizing inductance may resonate with 
the stray capacity of the circuit, producing ringing in the output. The 
initial amplitude depends on the amount of energy previously stored in 



a-l a 



t f 

*m i c Am 



196 ma 




2.22-*! 16.26 //sec 
i i 

200' m'a 



fct 



2.22 



'.) 16.26.Msec 
587 v 



21 vl 



18.48 ^sec 









191 v 


V* 




E a 



3v 






V 

















18.48 /tsec 


t 


E» 


-At 


W_ 








n 











E a - 

Piq. 12-4. Waveshapes produced in the inductively timed blocking oscillator. The 
values given are those found in Example 12-1. 

the transformer. If the oscillations damp out slowly, the next change in 
polarity may force the transistor back into its active region, resulting in a 
second, undesired output pulse — the circuit would become astable. 

Example 12-1. The blocking oscillator of Fig. 12-1 employs a transformer having 
L m = 2.5 mh, L. •» 100 fth, » = 3, and C, = 20 npi. The transistor is charac- 
terized by 



Tix = 20 ohms 
r c = 1 megohm 



or - 0.98 



turn-on time 0.1 /usee 
turn-off time 0. 1 ^sec 



364 switching [Chap. 12 

And the remaining parameters are E bb = 21 volts, E a = 3 volts, R = 500 ohms, 
Rl = 5 K, and C = 10 /if . We wish to construct the various waveshapes encountered 
(Fig. 12-4). 
From Eq. (12-3) the saturation time is given by 

*' = 22 (0.98 X 3 - 1)10" - 10 ~ Se ° 

but since the turn-on time is 0. 1 ^sec, this would be the proper value of h; the equation 
gives a solution that is a factor of 10~ 3 too small. 
The emitter current next rises to 

I.m = Ho( 2 ^ - 3) = 200 ma 

with the time constant t 2 = 0.556 jusec. At the end of this interval the collector 
current is only 

, 200 ma „_ 
lim = g — = 67 ma 

In the saturation region the magnetizing current starts increasing as 

20 , 



2.5 mh 



When this current builds up to the difference between <*/,„ and /,„, the transistor 
turns off. Under the conditions of this problem, the transformer must tolerate 
129 ma of magnetizing current without saturating. Thus t, is given by 



[0.98(200) - 67J10-' = (|| X 10') J 



or t% = 16.26 iiaec. 

The peak voltage developed across the primary is 

Eim - (196 - 67)10-» X 9 X 500 - 566 volts 

Unless limited by the judicious use of diodes, as shown in Fig. 12-5, this extremely 
large voltage will destroy the transistor. The voltage at the emitter winding is only 
Eim/n, or 

E lm = 56 % = 188 volts 

Recovery is with the relatively fast time constant 

2.5 X 10-« 



9 X 500 



= 0.54 /isec 



Figure 12-5 shows how to prevent the saturation of the transistor as 
well as how to limit the excessive amplitude backswing. Zener diode 
D s fires before the transistor can saturate and so sets a bottoming voltage 
of Es volts. The saturation conditions are transferred from the collector 
diode to the Zener diode. Except that the primary voltage builds up 
to Eu, — E a volts instead of to E M , the solution of the circuit is identical 
with that previously discussed. Diode Db conducts on the backswing 
if the terminal voltage exceeds E B . Since it shunts the coil with its 



Sec. 12-3] 



THE BLOCKING OSCILLATOR 



365 



low forward resistance, the discharge time constant would be greatly 
increased. In the solution of the recovery interval, two time constants 
must be considered, the long one holding while Db conducts and the fast 
one (Eq. 12-11) holding after the voltage decays below Eb. 




t 3 
Fig. 12-5. Zener-diode limiting of the collector voltage of the blocking oscillator. 

12-3. Transformer Core Properties and Saturation. The inductance 
of the blocking-oscillator transformer is proportional to the ratio of flux 
linkages and the current establishing this flux. 



N ^ 10- 8 
at 



henrys 



(12-12) 



At low frequencies or even at d-c, the relationship between the flux 
and the excitation is adequately expressed by the hysteresis curve (Fig. 
12-6). Assuming ideal conditions, B is simply the flux per unit area and 
H is proportional to the ampere-turns per unit length. For large signals, 
B = pH, and hence the inductance is proportional to the permeability 
of the core. From the hysteresis loop of Fig. 12-6o, we can see that ix 
varies over rather wide limits as the core flux increases from zero to 
saturation. 

If the hysteresis loop is approximated by the rectangular plot of Fig. 
12-6b, we avoid the necessity of treating a circuit containing a continu- 
ously changing inductance. After lumping the core losses together with 
the external resistance, it can be assumed that 



and 



L=L m 



M < N 
Itl > \i.\ 



366 



SWITCHING 



[Chap. 12 



where i, is the value of the coil current corresponding to the saturation 
magnetizing force H,. The inductance in the saturation region approxi- 
mates that of the coil in air; since L, <£. L m , it is often taken as zero. 

Because it requires finite time for the applied excitation field H to 
propagate into the core and establish and align the randomly oriented 
magnetic domains, the above formulation is not completely correct. 
With a pulse excitation, the flux is initially only at the surface, and con- 
sequently the effective permeability is quite low. It increases as the field 
penetrates into the core. Even if the excitation is well above the satura- 
tion value, it may still take several microseconds to saturate a thin core. 
The penetration time may be long compared with the width of the pulse 
generated in the blocking oscillator, and consequently the assumption 
of a constant L m is not valid. 





(a) (b) 

Fig. 12-6. Hysteresis curve and piecewise-linear representation. 

In order to avoid this inconsistency, we would have to postulate a 
magnetizing inductance whose value increases with time from slightly 
above that of an air-core coil to a peak given by the steady-state core 
characteristics. A circuit element which exhibits a response of this 
nature is impossible to use in any simple approach to circuit analysis. We 
shall therefore continue to explain the blocking-oscillator operation on the 
basis of a fixed L m and accept the resulting errors. However, the accu- 
racy of calculations can be improved if the inductance value used is not 
that at direct current but rather that measured under pulsed conditions. 

To summarize: 

1. For pulse durations greater than several microseconds, the assump- 
tion of a constant L m gives reasonably correot times. 

2. When generating pulses of less than 1 or 2 ^sec, the duration cannot 
be accurately calculated from any simple model. If a constant L„ is 
assumed, then the actual pulse will terminate sooner than calculated. 

3. A finite time is required to saturate the core, and therefore this 
problem need be considered only when generating wide pulses. 

Returning to the blocking oscillator, we shall now examine the opera- 
tion when the core does saturate. Up to this point, the response will be 



Sec. 12-4] the blocking oscillator 367 

identical with that discussed in Sec. 12-2. Since point C of Fig. 12-3 
indicated the start of the magnetizing-current build-up, the saturation- 
current point D' would fall somewhere along line segment CD; for a 
high-permeability core, it might be quite close to the initial point. This 
is the termination point of the pulse; the lower the saturation value, the 
more it is foreshortened. 

Once the transformer saturates, L m drops to a very small value. The 
primary current increases rapidly, limited only by the total series resist- 
ance. At the same time, the secondary voltage falls, the emitter current 
drops, and the transistor again becomes active. As the stored energy 
dissipates, the left-hand branch of the hysteresis curve is traversed by the 
operating point. As a result, a large reverse voltage pulse appears at 
the collector. The time required for the decay depends on the nature 
of the transformer core. (This problem is treated in more detail in Chap. 
13.) Because of the extremely nonlinear behavior, it is difficult to define 
the decay time in terms of the specific circuit parameters. In general, 
the manufacturer's specifications as to the rise and fall times and pulse 
width would be used in the circuit design. 

12-4. Capacitively Timed Blocking Oscillator. The coupling capac- 
itor C is now reduced in size until it alone will control the pulse dura- 
tion. Instead of the increasing collector current bringing the transistor 
out of saturation, the decreasing emitter current, due to the charging 
of C, will now so serve. To simplify the following analysis, we shall 
assume, quite incorrectly, that the magnetizing current remains zero 
over the complete pulse interval. Moreover, since the On time constant 
is on the same order of magnitude as LJnhw, the approximation that the 
initial build-up is complete before the timing element starts charging 
is no longer completely valid. We shall, however, assume that the peak 
values expressed in Eqs. (12-4) and (12-5) are still correct and shall 
begin our analysis from this point. By doing so, the nature of the 
reduction in the pulse duration may be seen most readily. 

From the model of Fig. 12-7o, we can see that C charges from its 
initial value of E a toward the TheVenin equivalent voltage 

The time constant is 

r 3 = C(R || r„) » Cm 

During this interval, the emitter current decays from its peak[Eq. (12-4)] 



I em — ( E a ] 

m\n ) 



368 SWITCHING 

toward the steady-state value of 

E. 



[Chap. 12 



/.(«>) = - 



r« + R — R 



The collector current will always be 1/n times as large. Consequently, 
because the transistor starts out saturated, it will remain so until I, 
falls to zero. 




Fig. 12-7. Models holding for the capacitively timed blocking oscillator, (a) Model 
holding during the charging interval; (6) model defining the recovery period. 

The duration of the unstable state, as found from the exponential 
charging equation, is given by 



T3 



In 



r.(«>) - I. 



or by substituting the appropriate terms, 

^^[^(iF.- 1 )] (12 " 14) 

If R 5J> rn, this portion of the pulse will consist of virtually the complete 
exponential. To avoid the ambiguity arising when the switching point 
is close to the steady-state value, E a might be made highly negative and 
R might be reduced. 

At the termination of the pulse, the transistor switches off and the 
recovery model reduces to the one shown in Fig. 12-76. Because the 
approximations made in the course of the above analysis neglected the 
energy storage in the transformer, the exact nature of the response over 
this portion of the cycle is somewhat difficult to define. One type of 
response would be expected when the secondary magnetizing inductance 



Sec. 12-4] 



THE BLOCKING OSCILLATOB 



369 



limits the rate of capacitor current flow, and yet another type where the 
transformer saturates rapidly. Regardless of the particular controlling 
element, the energy storage will almost always produce a large spike of 
voltage, and then either ringing (which must be suppressed) or an 
exponential decay back toward the original state. Waveshapes appear- 
ing at all elements, illustrating the type of recovery to be expected, 
appear in Fig. 12-8. 






f*-3.8,«sec-»-i 


t 


t 




_L_jw 




1 i/v 




\ V 


t 


Iem r U 


_____\J 









Fig. 12-8. Waveshapes appearing in the capacitively timed blocking oscillator. The 
recovery region illustrates ringing. 

We shall now consider the change in the response of the blocking 
oscillator of Example 12-1 when C is reduced from 10 to 0.02 /if. From 
Eq. (12-14), the duration of the final portion of transistor saturation 
lasts for 



h = Cn 



■-[ 



1 + W(7 

t- 40 V 



1) 



Since this is almost the complete exponential, 



; 4CVxi = 4(0.02 X 10-o) X 20 = 1.6 M sec 



370 



SWITCHING 



[Chap. 12 



K»m 



The pulse width, under this condition, is approximately one-tenth the 
time calculated when L m controlled the timing. 

We should note that the maximum pulse duration is due to the current 
build-up in the magnetizing inductance. Either the charging of C or 
the saturation of the core can only reduce this time. Furthermore, the 
simultaneous magnetic-flux build-up means that the peak currents that 
are reached and the actual pulse duration will always be less than calcu- 
lated on the basis of the idealizations made above. C is almost never 
used to control the pulse width, but it is used to time the interval between 
pulses in a free-running blocking oscillator (Sec. 12-5). 

12-5. Astable Operation. As might be anticipated from the previous 
discussion of monostable and astable operation in Chaps. 10 and 11, 

shifting the quiescent operating point 
from the cutoff to the active region will 
make the blocking oscillator free-running. 
For example, the astable circuit of Fig. 
12-9, except for the reversal of the 
emitter battery, is identical with the 
monostable circuit of Fig. 12-1. 
■ During the pulse interval, the large 
emitter current charges C slightly posi- 
tive. After the circuit switches off, 
then the accumulated charge maintains 
the emitter back-biased. Eventually 
the voltage from emitter to ground de- 
cays to zero and the circuit generates another pulse. 

At the instant of switching on, the net charge in C must be zero. The 
voltage at the transformer secondary has not as yet built up to E^/n, 
and therefore the emitter state depends solely on the voltage across C. 
If this voltage were positive, the emitter would be back-biased, and 
if it were negative, the circuit would have switched on earlier. Except 
for the change in the initial condition, the operation of the blocking 
oscillator, during the pulse interval, is identical with that discussed in 
Sees. 12-2 and 12-3. The peak negative voltage across the emitter and 
the peak current flow are now given by 

*»=-?* I- = J^ (12-15) 

For the purposes of computation we shall assume that this current 
remains constant over the complete pulse interval. Since 7 em flows 
through C, it will produce a net voltage change of 




Fig. 12-9. An astable blocking os- 
cillator — voltages shown for the 
saturated region. 



A£ = -q I., 



M 



(12-16) 



Sec. 12-5] 



THE BLOCKING OSCILLATOR 



371 



where At is the width of the pulse [At = f s + h as defined in Eqs. (12-6) 
and (12-9)]. The capacitor thus charges linearly from its initial value 
of zero to 

J5«-(A0 = +AJS . 

as shown in Fig. 12-10. In order to sustain a constant emitter current 
over the pulse interval, the change in the capacitor voltage must be small 
in comparison with the voltage across the transformer secondary. 



|#„»|« ; 



(12-17) 



Combining Eqs. (12-15), (12-16), and the inequality of Eq. (12-17), we 
can express one of the limiting conditions as 



C» 






(12-18) 



At the end of the pulse interval the transistor is driven hard off by 
the inductive backswing. After the initial recovery with the fast time 




Fig. 12-10. Emitter and capacitor voltage waveshapes in the astable blocking oscillator. 

constant, r 6 = L m /n*R, the charge in C will maintain the cutoff condition. 
As it discharges toward — E a , with the relatively long time constant 
r = RC, the emitter voltage decays from +E gm to zero. Thus the dura- 
tion of the interval between adjacent pulses is given by 



RC\n 



-Eg — E ql 

~E a 



(12-19) 



And, for a given period, Eq. (12-19) determines the size of R. 

The astable blocking oscillator is commonly used to generate trains 
of narrow pulses having relatively wide separation. The time defined 



372 switching [Chap. 12 

in Eq. (12-19) very nearly represents the complete period of the generated 
signal. 

As with all free-running circuits, the blocking oscillator may be 
synchronized to an external control signal. It must, of course, have 
a frequency and amplitude placing it within the regions of synchroniza- 
tion. Because the pulse is so very narrow when compared with the full 
period, the solution of the synchronization problem would be much 
closer to that found for the sawtooth sweep in Chap. 5 than to any regions 
which might be constructed for a multivibrator. 

Example 12-2. In this problem we wish to calculate the values of C and R neces- 
sary to establish a pulse spacing of 2.5 msec. The blocking oscillator generates a 
5-tiaec pulse. The significant parameters for the recovery interval are r n =40 ohms, 
En, - 20 volts, E„ = 1 volt, n = 2, and L m = 2.5 mh. 

From the above parameters, Eq. (12-18) yields 

C» 5X 4 *°~' = 0.125 „f 

In order to keep its size within bounds, we shall choose C = 2 /if . The peak voltage 
developed across it is 

E, m = ± — At = 0.625 volt 
C nru 

The remaining circuit component R is now found from Eq. (12-19). 

R ' C ** + *- = 2X 2 l 5 0-^ln "l' 62 5 " 2 ' 580ohm " 

Ea 

As a final step we should verify that the voltage backswing due to the energy stored 
in the transformer will have damped out long before the transistor turns back on. 
This time constant is 

L„ 2.5 X 10-' _ n „, 9 „, 

Tt = JfiE " 4 X 2.58 X 10' " ° 242 " 8e0 

and thus the backswing occupies an insignificant portion of the recovery interval. If 
a Zener diode was included, this might no longer be true. 

12-6. A Vacuum-tube Blocking Oscillator. The circuit of Fig. 12-11 
is probably the most difficult to analyze of all those used for pulse 
generation. Many words have been written purporting to detail its 
operation, but none of the papers or texts have been completely success- 
ful. The difficulties faced are many: 

1. More than a single mode of energy storage is always operative. 

2. The various parasitics must be included; they establish the switch- 
ing time and thus place a lower limit on the pulse duration. 

3. The tube operates in a region not previously defined, where its 
parameters change drastically from the small-signal operation. In this 
region the values of fi, r p , and r e are not even well known. 



Sec. 12-6] 



THE BLOCKING OSCILLATOB 



373 



4. Because of the flux build-up and even possible core saturation, 
the timing inductance becomes a function of the magnetizing current 
and the pulse width (Sec. 12-3). 

Difficult problems always have a peculiar fascination. In discussing 
them, one tends to go deeper and deeper until perspective is lost and 
the subject appears overly important. In an effort to avoid this pitfall, 
the following arguments are concerned with two rather limited goals: 

1. The development of a tube model for the new region of operation 

2. The construction of a set of reasonably consistent models and the 
extraction of some essential information when these models are inade- 
quately defined 




100 K 



Ideal transformer 



-20 v 



Fig. 12-11. A vacuum-tube monostable blocking oscillator and its general equivalent 
model — timing controlled solely by the transformer. Waveshapes produced are 
shown in Fig. 12-16. 

Whenever the analysis becomes overly complicated or does not give 
promise of rewards commensurate with the work involved, it will immedi- 
ately be dropped. 

We shall assume that the reader has some knowledge of the waveshape 
produced (Fig. 12-16). Such an assumption is not unreasonable; the 
laboratory must always be a close adjunct to the paper work, especially 
when treating a complicated circuit. It is far easier to clear up fine 
points with a little laboratory work than from a model that is at best only 
an idealization. 

The first step in the analysis of the vacuum-tube blocking oscillator 
is to define a new tube model. This becomes necessary because as the 
tube in the circuit of Fig. 12-13 switches on, the large plate drop, coupled 
by the transformer, drives the grid beyond the point at which the previ- 
ously derived models hold. In fact, within the timing region, the grid 
voltage may even exceed that at the plate. When this happens, the grid 
loses control over the plate current. Any increase in the grid voltage 
increases the total cathode current up to the emission capabilities of the 
tube. But because the grid is at the highest potential, almost all the 



374 switching [Chap. 12 

additional current flows to it; the plate current remains essentially 
constant. As can be seen in Fig. 12-12a and b, when E e > Eb, the fam- 
ily of plate volt-ampere characteristics degenerates into a single curve. 
It follows directly that the plate-circuit response may be represented by a 
simple diode having the forward resistance r p , (Fig. 12-13). 

200 

175 

150 

1 125 

•^100 

o 

* 75 
50 
25 







X 


tP | 














A 


r 


S* 


■> 














/ 


/' 


^ 


i.'JPn 












ro 


</ 


/' 




^» ■*'£-• 


V2.0 








1 


's 


' ^ 










1-15 






















ZS- 


+5 

* 




<& 










~x 








=»^ 


"S" 












—40 



50 



100 150 

E b , volts 



200 



250 



*uu 
175 




y 




-*A 


i 














/ 





. 


.&> 


.•W 










150 
125 

^100 
75 




A 


- — 


— — 


.— *"' 


—+2 


5 










n 


— 


*-** 






-B*- +w 






"•V 














—— + 


.0 




















.+5 




-*> 


50 
25 


LBf ■ 




£ 


A 































-50 



£ w 50 



(6) 



100 150 

e b , volts 



200 



250 



Fig. 12-12. Vacuum-tube positive grid characteristics and piecewise-linear represen- 
tation (after 12AU7 characteristics). 

The value of the plate resistance in the absolute saturation region of 
Fig. 12-126 is given by 

Eh\ E c \ 



r " hi hi 



(12-20) 
But the plate voltage intercept of the extended active-region character- 



Sec. 12-6] the blocking oscillator 375 

istics through the point (En, hi) is 

Em — —pEci 

Thus, from the geometry, the plate resistance in the active region may be 
expressed as 

r, = *Lfl*? - <*+»*' (12-21) 

Comparing Eqs. (12-20) and (12-21) allows us to write the saturation 
resistance in terms of the active-region parameters. 

rm = ^J (12-22) 

The grid resistance used in the model of Fig. 12-13 must also be 
reduced to one-half or one-quarter of the value measured at the lower 
values of grid voltage. Furthermore, 

.... 8° I i op 

this resistance is extremely nonlinear. 
At very high values of voltage and 
current, the grid driving-point charac- 
teristic may even begin to approximate ■ ._ 

a constant voltage drop more closely Aft 

than a pure resistance. In the in- FlG - 12 " 13 - Triode model for the abso- 

terests of simplicity, however, we shall lute saturation Te & oa - 

assume a constant resistance whose value would be found in the vicinity 

of e„ S e b . 

For the purposes of discussion, the operation of the vacuum-tube 
blocking oscillator of Fig. 12-11 may be divided into the same four 
regions found for the transistor circuit. These are characterized by: 

I. Active region. The positive loop gain drives the tube into "abso- 
lute" saturation (that is, e c > e^). 

II. Tube saturation region. Here the circuit voltages buifd up toward 
their peak values. 

III. Timing. Current build-up in the transformer, core saturation, 
and/or the charging of C return the tube to the active region and ter- 
minate the pulse. 

IV. Switching and recovery. The regeneration of the circuit turns 
the tube off, and the energy previously stored must now be dissipated. 
The output consists of a large voltage overshoot and either ringing or an 
exponential decay toward the initial conditions. 

Of all the parasitic elements present, only the two of major significance, 
that is, L e and C„ will be included in the linearized models. The stray 
capacity, which is distributed throughout the circuit, may be lumped 
into a single element. But where should it be placed? If it is inserted 



376 switching [Chap. 12 

from the grid to cathode or directly across the magnetizing inductance, 
nothing constrains the plate voltage; once the tube turns on, the full 
supply voltage appears across L e and the plate immediately bottoms at 
zero. This will not occur in the physical circuit. If the model permits 
such a drop, the model is incorrect. To prevent the abrupt change in 
ej, the stray capacity should be connected from the plate to cathode as 
shown in Fig. 12-11. 

Without actually solving for the exact time response, we shall now 
discuss the basic behavior of the circuit and the role of the various 
parameters. 

Region I. Initial Rise. The voltage swept through during the 
initial portion of the grid rise, from cutoff to zero, is a very small per- 
centage of the total voltage change. Moreover, since the grid will not 
load the plate circuit, the loop gain will be much higher than that meas- 
ured in the positive grid region. The rate of voltage change is very rapid, 
and the time contribution to the over-all pulse duration insignificant. 

In the positive grid region, the nature of the response would be found 
from the model of Fig. 12-14a. Since 

ei 

e c = = — %inr c 

n 

the circuit poles would be given by the solution of the determinant 

R + w. Anr °~w. 



or 



, , / 1 , n*rA , R + w 2 r, + nAr e _ „ n „ 9 „, 

p* + (rc, + t;) p + — bljc. ° {12 " 23) 

For proper switching, one pole must lie in the right half plane. With the 
values given in Fig. 12-11, the two poles are located at 

Pl = -7.43 X 10 7 p 2 = 2.43 X 10 7 

In a relatively short time the effect of the negative exponential will 
have damped out. The time response is primarily due to the positive 
pole, and we may approximate the plate-voltage fall by 

From the definition that the rise time is the time required to charge 
from 10 to 90 per cent of the bounding voltage, 

fx = — = 0.091 /usee 
Pi 



Sec. 12-6] 



THE BLOCKING OSCILLATOR 



377 



The negative pole, along with the additional parasitic elements not 
included in the model, will slow the rise by a factor of 1.5 or 2. Any 
increase in L, or C, would also adversely affect the switching time. 








I'M'r 

Ebb nEcc 
(c) 

Fig. 12-14. Models holding for regions I and II of the vacuum-tube blocking oscillator 
of Fig. 12-11. (a) Incremental model holding within the active region; (6) model 
denning the rise in the saturation region; (c) model used to solve for ft,min and E c , m „. 

This portion of the response is finally bounded when the grid rise and 
plate fall drive the tube into saturation. Reflecting the grid circuit 
voltages into the transformer primary results in 

En + nE cc = e b + e L + ne c 

where ej, is the drop across the leakage inductance. Solving for 

e h = e e = Eb\ 



we obtain 



En = 



Ebb + nE cc — 6l 
n + 1 



(12-24) 



Equation (12-24) cannot be solved until e L is evaluated. But this is too 
difficult to do. The particular voltage at which the tube saturates is 
not crucial to the circuit operation. Because L e is small, the voltage 
developed across it will not be large even at the boundary. Arbitrarily 
assuming e L = 10 volts, the circuit used as an example yields 



Ebi = 



200 - 40 - 10 



= 50 volts 



Region II. Final Current Build-up. After the tube saturates, the 
circuit model reduces to the one shown in Fig. 12-146. Because R L 
is very much larger than the 285-ohm plate saturation resistance, the 
3,000-ohm external load may be neglected. The two poles in the left- 



378 SWITCHING 

hand plane are now located by the solution of 

r p . + n 2 r c 



or at 



p . + (_L_ + V^\ 

V + \r P .C + L e ) 
pi = -9.51 X 10 7 



P + 



P2 



r P .L.C, 



[Chap. 12 



(12-25) 



-1.75 X 10 7 



Charging continues from the boundary value of E e i = En toward the 
steady-state conditions, which can be found from the model of Fig. 
12-14c. The pole located closest to the origin is associated with the 
longest time response, and consequently the duration of this charging 
interval is approximately 



iPil 



= 0.228 Msec 



From Fig. 12-14c the peak voltages are given by 

Eh,aLi * = r , + n*r c ^ Ebb "*" ^ 
Ee.nuti = ; — -y— (Etb + nE^.) 



r ps + n*r c 



(12-26a) 
(12-266) 



Solving these equations in the circuit used as an example, we find that 
the plate falls to 20 volts while the grid rises to +70 volts. In an actual 
circuit r c would decrease markedly when the grid voltage greatly exceeds 
e t . This lowers E c , aiI and raises E htaia . 

C 



HI 



Fig. 12-15. Model holding over the timing region. 

Region III. Pulse Timing. This, the most important region, is 
denned by the very simple model of Fig. 12-15. Note that because 
L m » L e , C, is much less significant and may even be omitted. The 
magnetizing current begins increasing with the time constant 



n'r e 



toward the steady-state value 



/.. = 



Ew> 



Sec. 12-6] the blocking oscillator 379 

Meanwhile, the plate voltage rises from jE t , mill toward Eu, and the grid 
decays back toward E cc . The two charging equations are 

e„ = E», - (B» - E h , mia )e-'i* (12-27o) 

e„ = Ecc - (E C c - Ec.^Je-'i* (12-276) 

Eventually the decay at the grid and the rise at the plate will permit 
the tube to reenter the active region. Solving Eqs. (12-27o) and (12-27&), 
the time elapsed until ei, = Em = E c % is 

h = t In „ „ (12-28) 

Ebb ■+■ Jbcc 

The value of E ai = E c % may be found by substituting the U back into 
Eqs. (12-27) or by eliminating the exponential term between these two 
equations. The simplest expression for the boundary value becomes 

Ec2 = E i2 = Etb + *f" (12-29) 

We might observe that the 53 volts at which the circuit leaves the satura- 
tion region is slightly higher than the 50 volts at which it entered. The 
duration of the pulse is 

h = 20 X 10-* In (1 + 5 %2o) = 20 X 10~« In 1.227 = 4.08 j*sec 

At the end of the pulse the magnetizing current has reached a peak of 

/m(f 8 ) = — (1 - e-"") = 130 ma 

Tp. 

If the transformer cannot tolerate such a current without saturating, 
the pulse would terminate proportionately earlier. 

In practical blocking oscillators, the high grid current which charges C 
increases the rate of the grid voltage decay. Switching will occur 
somewhat sooner than given by Eq. (12-29). This same charging is 
also used in the astable circuit to back-bias the tube during the interpulse 
period. 

Region IV. Termination of the Pulse. Once the circuit again becomes 
active, regeneration drives the tube toward and even beyond cutoff. 
As long as the tube remains active, the poles defining the response will be 
given by Eq. (12-23). Below cutoff the time response is defined by the 
parallel Rl, L m> and C, circuit. Depending on the degree of damping, 
the output may be either an exponential decay or a damped sinusoid. 
The final decay is from a peak determined by the energy previously 
stored in L„. The external damping Rl not only limits the backswing 
to a safe value, but also prevents the ringing that may cause a false 
retriggering of the circuit. 



380 



SWITCHING 



[Chap. 12 



The magnetizing current of 130 ma flows through Rl after the tube is 
cut off. This results in a peak voltage at the plate of 

#w = Eu, + I m (h)R L = 590 volts 

At the grid the maximum possible backswing is 

£ .„i. = -20 - 39% = -215 volts 

The above calculation neglected the stray capacity, which will, of 
course, slow the switching and reduce the amplitude of the backswing. 




Fig. 12-16. Plate and grid wavi 
Kg. 12-11. 



for the monostable blocking oscillator of 



12-7. Some Concluding Remarks. Even a small receiving tube, 
limited to 2 or 3 watts plate dissipation and 0.5 watt grid dissipation, 
is able to supply 50 or 100 watts of peak power over the narrow pulse 
interval. To ensure that the average power will not exceed the ratings, 
the blocking oscillator is normally operated with a duty cycle of 1 to 
5 per cent (time on as compared with the total period). For pulses hav- 



Sec. 12-7] 



THE BLOCKING OSCILLATOR 



381 



ing a larger duty cycle, multivibrators would be used in preference to the 
blocking oscillator. 

On the other hand, only a small amount of power is dissipated by 
the saturated transistor during the pulse. If large currents are switched 
in the collector circuit, then the peak collector dissipation is reached 
within the active region — during the turn-on and turn-off times. Larger 
duty cycles are possible without damaging the transistor. Usually the 
resistance inserted to limit the voltage backswing imposes much more 
severe restrictions on the duty cycle (long recovery time) than do the 
power considerations. 

Of the many other modes of blocking-oscillator operation possible, 
one of the most common is where the transformer primary is designed 
to resonate with the stray capacity. The circuit operates as a tightly 
coupled tuned plate oscillator such as discussed in Chap. 14. This 
circuit is not really a relaxation oscillator, but it can still be analyzed 
by the methods discussed in this chapter, with, however, the active-region 
model defining the major portion of the response. The waveshapes 
appearing at the plate and grid would be portions of a sinusoid, although 
somewhat distorted by the changing grid resistance. 



PROBLEMS 

12-1. In Example 12-1 the stray capacity is increased until the circuit is on the 
verge of becoming oscillatory during recovery. 

(a) Calculate the minimum value of C s which will resonate with L„. 

(6) When C, is ten times the minimum value, sketch the response in the recovery 
region. Pay particular attention to any retriggering of the blocking oscillator. 

12-2. Figure 12-17 shows an emitter-collector-coupled blocking oscillator where 
the period is independent of any power-supply variations. Except for Ri and Ri, all 
components are as given in Example 12-1. 




Fio. 12-17 



382 



SWITCHING 



tCHAP. 12 



(a) Prove or disprove the above statement. 

(6) iJi is 500 ohms, Ri is adjustable, and C 2 is very large. Plot the peak value of 
emitter current and the pulse duration as functions of ijj. At what value will the 
circuit stop operating? 

12-3. Figure 12-18 shows a collector-base-coupled blocking oscillator with the 
external load connected to a tertiary winding. 

(a) What is the minimum value of n for guaranteed operation? Indicate the 
polarity of the transformer connections. 

(6) Assume that the transformer turns ratio is two times the bounding value. In 
addition, the ratio L m /L, = 25:1. Specify the transformer parameters for a 5-Msec 
pulse duration. 

(c) Sketch and label the load current. 

(d) Compare the advantages and disadvantages of this configuration with the cir- 
cuit of Fig. 12-1. 

12-4. Draw the circuit for an emitter-base-coupled blocking oscillator. Using the 
parameters of Fig. 12-18 with leakage and magnetizing inductance (referred to the 
emitter winding) of L m — 10 mh, L, = 0.5 mh, and n = 3 (base winding), calculate 
the pulse duration. Sketch and label the load current and the collector voltage 
waveshapes. 



9+20 v 




C-10/if 
rii=40D 
/3-50 
A- 1,000 



-lv 



Fig. 12-18 



12-6. The blocking oscillator of Example 12-1 is modified to the circuit of Fig. 12-5. 
Repeat the waveshape calculations if the bottoming value is 2 volts. The Zener diode 
chosen to limit the collector voltage conducts at 40 volts; in series with it, we insert a 
recovery resistance to limit the peak at the collector to 75 volts. How would the 
circuit respond if the Zener resistance is 10 ohms? ohms? 

12-6. Show three methods of triggering the blocking oscillator of Example 12-1. 
Use both current and voltage input pulses. Specify the pulse height, polarity, and 
order of magnitude of the source impedance. Does any one method chosen have 
overwhelming advantages? Explain. 

12-7. The transformer of Example 12-1 is changed to one that saturates at 50 ma. 
Assume that L m drops from 2.5 mh to 250 jm after saturation. Plot the new wave- 
shape, paying particular attention to operation after saturation. Are all the assump- 
tions made in the text still valid? What happens after the transistor turns off? 

12-8. Figure 12-19 shows a blocking oscillator designed so that the pulse width 
can be controlled by a constant current injected into a tertiary winding. This cur- 
rent establishes a bias flux and thus influences the saturation point. Saturation 



THE BLOCKING OSCILLATOR 



383 



occurs at a field strength of 50iV X 10~* amp-turns, where N is the number of turns in 
the collector winding. Specify the parameters (L m , C, and B) that will permit the 
adjustment of the pulse duration from 5 to 50 psec. Neglect the effect of L, on the 
pulse width. 



<?+20v 



-0.98 




ri'i_40Q ci 



12-19 



12-9. (a) The circuit of Fig. 12-18 is made astable by reversing the base bias bat- 
tery. In addition, L„ = 5 mh and n = 0.2. Calculate the value of C that will set 
the period at 6 msec. Neglect L, in your calculations. Show the recovery portion 
of the collector waveshape and the capacitor current. 

(6) By how much would the charging current reduce the pulse duration? Make 
all reasonable approximations in your calculations. 

12-10. Repeat the calculations for the vacuum-tube oscillator discussed in the 
text when C, = 0. Sketch the waveshapes and compare the times with those pre- 
viously found. Plot the path of operation of these two cases on the volt-ampere 
characteristic and compare the location of significant points. 

12-11. Can eh in Eq. (12-25) be evaluated from the known conditions of the prob- 
lem? If it can be, do so. If it cannot be, explain why not. 

12-12. (a) Justify the approximations made in finding h and t t in the text in 
Sec. 12-6. 

(6) If these simplifications were not made, would it be possible to calculate the 
exact response? Explain. 

12-13. (a) Find the value of Bl, in terms of the other circuit parameters, that will 
just prevent ringing in the recovery region (Fig. 12-11). 

(6) Evaluate this resistance for the circuit discussed in the text. Calculate the 
backswing produced. 

12-14. Sketch and label the plate-current, grid-current, and cathode-current wave- 
shapes for the blocking oscillator of Sec. 12-6. Calculate the power dissipated in 
the tube over the pulse interval. 

12-15. Plot the peak grid voltage; the minimum plate voltage; the peak grid, plate, 
and cathode currents; and the pulse duration as a function of the transformer turns 
ratio. Let 0.2 < n < 5 and use the example given in the text as your circuit. The 
initial rise may be neglected in your calculations. Can any conclusions be drawn as 
to the "best" transformer? 

12-16. An astable vacuum-tube blocking oscillator is shown in Fig. 12-20. Calcu- 
late the pulse duration and the period. Sketch and label 2 full cycles of the grid 
voltage waveshape. 



384 



100 K 



SWITCHING 

?+200v 



[Chap. 12 




3.3 K 



£ m -25 mh 
L e -500pb 
CSOjyif 



Fig. 12-20 



BIBLIOGRAPHY 

Benjamin, R.: Blocking Oscillators, /. IEE (London), pt. IIIA, vol. 93, pp. 1159-1175, 
1946. 

The Blocking Oscillator, Wireless World, vol. 63, pp. 285-289, June, 1957. 

Linvill, J. G., and R. H. Mattson : Junction Transistor Blocking Oscillator, Proc. IRE, 
vol. 43, no. 11, pp. 1632-1639, 1955. 

Narud, J. A., and M. R. Aaron: Analysis and Design of a Transistor Blocking Oscil- 
lator Including Inherent Nonlinearities, Bell System Tech. J., vol. 38, no. 3, 
pp. 785-852, 1959. 



PART 4 
MEMORY 



CHAPTER 13 

MAGNETIC AND DIELECTRIC DEVICES AS MEMORY 
AND SWITCHING ELEMENTS 



In this chapter we shall be making use of just those properties of 
magnetic and dielectric materials that were so annoying in linear circuits. 
One of the more important is hysteresis, i.e., the dependency of the 
present state on the past excitation, the material's memory. The other 
characteristic, saturation, which allows several regions of operation, 
permits the use of these solid-state devices in controlled gates and other 
circuits previously restricted to diodes, tubes, and transistors. Not only 
are special materials used, but these are also treated to exaggerate the 
hysteresis and saturation regions before being fabricated into cores or 
capacitors. The behavior of magnetic materials has been studied much 
more extensively, and consequently there is a much wider range of 
properties available than can be found in the dielectric mediums. For 
this reason we shall concentrate mainly on those circuits making use of 
ferromagnetic devices. However, as this situation will most probably be 
rectified in the near future, the last sections of this chapter will consider 
certain ferroelectric devices and circuits. 

13-1. Hysteresis — Characteristics of Memory. We must sometimes 
store the information contained in a sequence of pulses which are gener- 
ated by a multivibrator or blocking oscillator and transmitted through 
various gates. In digital computers, rapid storage and read-out are 
essential for the solution of complex problems. Besides storing data and 
answers, the program controlling the sequence of the solution is fed into 
the memory banks. As each step is completed, the computer switches 
to the next memory location for further instructions. 

Storage consists in switching a bistable device from its low to its high 
state, arbitrarily designated by and 1. Thus a time sequence of pulses 
may contain almost any desired coded information. If a pulse is present, 
it would be indicated by 1, its absence noted by 0. For example, one 
particular five-digit word might be 10011. Each digit contains one bit 
of information, and each would have to be registered in a separate mem- 
ory location. Upon the application of the appropriate read-out pulses, 
the excited memory units would switch from 1 to 0, with the previously 

387 



388 



MEMORY 



[Chap. 13 



stored information appearing as output pulses. Those in the state 
remain undisturbed. 

We shall now consider the particular characteristics required in the 
individual memory unit. 

1. It should have at least two definitive, widely separated levels. 
Once excited from the lower to the higher, the memory element should 
maintain the new state after the initial storage pulse has disappeared. 
This precludes use of all nonlinear devices having continuous character- 
istics. We cannot identify the state of the diode (curve a in Fig. 13-1) 
by looking at the zero excitation point. 




, 


1 
















(a) (b) (c) 

Fig. 13-1. Three typical characteristics of bistate elements. 

2. Both the process of storage and read-out should require a finite 
amount of energy. If the device switches levels with an absolute mini- 
mum input, what is to prevent it from switching erratically in the presence 
of noise? The two states of curve b in Fig. 13-1 are interchangeable at 
zero excitation; a device having this characteristic would be unreliable 
for use as a memory element. 

3. At any instant the state of the device must represent the past 
history of the system as well as the present excitation. 

4. In general, storage and read-out should require different types of. 
excitation. This is necessary because we do not want the memory to 
shift from 1 to when a storage pulse is applied or to store a read-out 
pulse falsely by switching from to 1. Such differentiation might be as 
simple as using positive and negative pulses for storage and read-out, 
respectively. 

It follows directly from the above statements that hysteresis is essen- 
tial in any device used for information storage (curve c in Fig. 13-1). 
It can be a natural property of the device, as in the ferromagnetic 
cores or in various ferroelectric dielectrics, or it can be simulated by the 
use of diodes and energy-storage elements, as discussed in Sec. 2-9. 
Bistable multivibrators, constructed either with tubes or transistors or 
with the two-terminal negative-resistance devices of Chap. 11, also exhibit 



Sec. 13-2] 



MAGNETIC AND DIELECTRIC DEVICES 



389 



hysteresis, and as we know, the state of these circuits is controlled by 
input triggers. 

As an example, consider the multivibrator of Fig. 13-2. Assuming 
that it is initially in the state (T 2 off and Ti on), the voltage at the 
grid of Ti is positive, limited primarily by the resistive loading of the 
conducting grid. R x , R z , and the positive grid resistance form a resistive 
summing network. Ti cannot be turned off and T 2 on until the external 
input ei is negative enough to bring the grid voltage to below zero, i.e., 
to bring the operating point to point A in Fig. 13-2. Regeneration then 
causes the jump to point B. Even if ei is made more negative, it will 
have no further influence over e 2 . When the input trigger is removed, the 
output finally stabilizes at point 1. The right-hand portion of the 






En 


*2 
D 




, 


B 


1 


C 

> 



(b) 

Fig. 13-2. Bistable multivibrator and input-output transfer characteristic illustrating 
hysteresis. 

hysteresis curve is traced upon the application of a positive read-out 
pulse, which resets the multivibrator to state 0. 

Among the disadvantages of multivibrators for use as memory ele- 
ments is their large size. Even when using transistors, the sheer bulk 
of the units needed to store 10,000 or 20,000 bits of information would 
force us to turn to some other storage device. Furthermore, since each 
multivibrator always has one element conducting, a large bank of them 
requires an excessive amount of power. Any failure in the power supply 
would automatically erase the stored information. For the above rea- 
sons, the use of multivibrators is restricted to systems requiring only a 
small memory capacity and to subsidiary registers in the larger computers. 

13-2. Ferromagnetic Properties. In order to explain, even cursorily, 
the salient features of the B-H curve of Fig. 13-3, we shall start with a 
microscopic region and increase our range until it encompasses the whole 
core. There exist, within the material, small domains where all the 
atomic magnetic moments are aligned in some preferred direction. At 



390 



MEMORY 



[Chap. 13 



room temperature iron has six and nickel has eight easy directions of 
magnetization which are related to the crystal structure. Separating 
the domains are walls composed of atoms whose moments are skew to the 
principal direction. Since, from a macroscopic viewpoint, the dis- 
tribution of magnetic moments is random, the material is normally 
unmagnetized. 

An external field increases the area of those domains magnetized in its 
direction and causes the contraction of those that have other magnetic 
moments. As the excitation is increased, portions of the walls become 



j Slope -Amax 




Fig. 13-3. Portion of the magnetization curve, showing some of the important con- 
stants of the material. 



unstable, jumping to the direction of forced orientation. The finite field 
required to produce this change results in the hysteresis shown in Fig. 
13-3. When all the atomic moments are essentially in alignment, we say 
that the material is saturated; it then supports a magnetic flux density 
B,. 

The presence of the external field is necessary to maintain the align- 
ment of the magnetic domains. After the excitation is removed, they 
relax to the nearest easy direction of magnetization. Instead of the 
original random orientation, there now exists a component of flux in the 
direction of the applied field, and this is called the remanence B r . To 
completely demagnetize the material, we must apply a reverse field that 
will overcome the internal coercive force H c . 



Sec. 13-2] 



MAGNETIC AND DIELECTRIC DEVICES 



391 



One important factor used to relate the field and flux is the magnetic 
permeability, 

, = & (13-D 

which is not a constant, but increases from a low value near the origin 
to a maximum near the peak of the magnetization curve, decreasing 
as the core finally saturates. Even though B„ B r , H c , and cr (the volume 
conductivity) are also needed for a complete description, **»„ serves as a 
convenient measure of the magnetic quality. Those materials having a 
high permeability almost always have a narrow hysteresis loop. For 
example, iron, which has a maximum permeability of 5,000, requires a 




Fig. 13-4. Typical hysteresis curves and equivalent static piecewise-linear represen- 
tation (4-79 Mo Permalloy). 

magnetizing force of 3.2 oersteds to develop the saturation flux density 
of 16,000 gauss. A high-permeability alloy, such as Permalloy (79 per 
cent nickel, 17 per cent iron, and 4 per cent molybdenum), may have a 
maximum m of 100,000 or higher. This material would saturate at a 
flux density of close to 8,000 gauss, when the field strength is approxi- 
mately 0.1 oersted. It has a coercive force of 0.05 oersted and a reten- 
tivity of 6,000 gauss. Note that the hysteresis loop of this second 
sample (Fig. 13-4) begins to approach a rectangle, so much so that it 
becomes valid to approximate the curve in the piecewise-linear manner 
shown. 

The above discussion was based on the response either to direct current 
or to relatively low frequencies where the flux in the core has time to 
build up to the maximum value permissible with the applied excitation. 
If the dynamic hysteresis loop were to be plotted by using a very-high- 
frequency excitation signal, then the area of loop, which represents the 



392 



memory [Chap. 13 

This effect is noticeable even at 



kB 



~H C 



H c 



H 



losses, would be greatly increased. 
400 cycles (Fig. 13-4). 

In pulse applications we are interested, not in the steady state, but in 
the much more complicated transient B-H response. With any suddenly 
applied magnetization force H a , we should expect an immediate flux 
density B a . It would not be present. The H field must produce the 
alignment of the magnetic domains before the flux given by the B-H 
curve can be sustained. In addition, the H field propagates from 
the surface into the core with a finite velocity. Initially, only the 
surface layer contributes, and therefore the effective y. is an extremely 
small percentage of the final value. It will, of course, increase as the flux 
builds up toward steady state. 

Both the coercive force and the retentive field are structural proper- 
ties, depending on such factors as grain orientation and microscopic 
imperfections, i.e., voids and inclusions. If, in working the magnetic 

alloy, the grain size is controlled and 
aligned to favor one easy direction of mag- 
netization, (!„„. can be greatly increased. 
One method used to produce almost square 
loop materials (Fig. 13-5) is to cool the 
hot-rolled sheet slowly in a hydrogen 
atmosphere with the magnetic field applied 
in the direction of rolling. The high per- 
meability is measured only in the direction 
of grain orientation. To obtain the full 
advantage, the thin tapes produced are 
wound into toroidal cores. Deltamax (50 
per cent nickel and 50 per cent iron), 
which has a maximum permeability of 70 000, has a squareness factor 
(B T /B.) of 0.98. 

Energy must be supplied to the excitation coil to overcome the losses 
involved in establishing and aligning the magnetic domains. The input 
per unit volume required to switch between two points on the hysteresis 
loop is given by 

w = (£*' H dB (13-2) 

The terminal characteristics of the coil depend on the amount of stored, 
and hence recoverable, energy. In switching from point 1 to 2 in Fig. 
13-3, the net energy supplied is proportional to the area between branch 
I and the B axis. Once the excitation is removed, the operating point 
traverses segment II from point 2 to 3. Only the energy represented by 
the small area between branch II and the B axis is recoverable. The 
area of the hysteresis loop lying in the first quadrant represents the 



Fig. 13-5. Square hysteresis loop 
such as found in a tape-wound 
toroidal core of Deltamax. 



MAGNETIC AND DIELECTRIC DEVICES 



393 



Sec. 13-3] 

dissipated power, and consequently the resistive component of the ter- 
minal characteristic. 

In square-loop materials (Fig. 13-5), all the supplied energy is dissi- 
pated and none can be recovered. We thus interpret the driving-point 
characteristic of the core as that of a nonlinear resistance. Since this 
case is of great interest, it will be considered in some detail below. 

Ferrites. Some of the disadvantages of the thin-strip cores have been 
overcome by the use of various molded ceramic semiconductors. These 
materials, such as NiOFeuO, (nickel ferrite) or MgOFe 2 3 (magnesium 
ferrite), have a much lower electric conductivity and consequently lower 
losses. The nature of the material is such that the domain switching 
time would be much less than that of the tape cores. Even though the 
maximum permeability is small (m„« = 1,000), it is almost independent 
of frequency. The easily fabricated square-loop ferrite cores, of almost 
any desired configuration, are widely used in high-speed memory arrays 
and switching circuits. 

13-3. Terminal Response of Cores. Before we can make use of cores 
in specific circuits, we must know their terminal response to various 
excitations. We have seen that 
this is a function of both the core 
geometry and the shape of the 
hysteresis loop. Because the gen- 
eral problem is quite complex, our 
discussion will concentrate on a 
square-loop core of relatively simple 
geometry. The core of Fig. 13-6 
is constructed by winding a thin 
tape of grain-oriented magnetic 
material on a bobbin. By making 
the following assumptions the sub- 
sequent analysis will be greatly 
simplified. 

1. At every point the radius of curvature is large compared with the 
thickness d. In any small region the core may be treated as an infinitely 
long plane sheet and the geometry reduces to two dimensions. 

2. The width h is much greater than the thickness d. Fringe effects 
may be ignored. Consequently the excitation can be assumed to be a 
current sheet flowing on the outer and inner surfaces of the core. 

3. The hysteresis loop is perfectly rectangular. It follows that the 
core, which is initially magnetized to — B„ switches to +B, as the H 
field propagates from the surface toward the center. 

4. Because of the symmetry, the field will penetrate at equal rates 
from both the inside and outside of the core. 




Fig. 



m 

13-6. Tape-wound core. 



394 



MEMORY 



[Chap. 13 



Once the magnetic field is established on the surface of the core, it 

creates the domain wall, which 
propagates toward the center, 
dividing the region having a flux 
density of +B, from that at — B.. 
Throughout the complete material, 
the flux will have one of these two 
values. Only across the domain 
wall will there be any change of 
flux with respect to either time or 
space. Figure 13-7 illustrates the 
conditions in the core before the 
field penetrates to the center and 
completes the switch in the satura- 
tion direction. 

Response to a Known Voltage 
Excitation. The terminal voltage 




Fig. 13-7. Cross section of the core illus- 
trating the flux penetration and the 
direction of the current flow. 



response of the excitation coil is given by 

«,(<) = N^ = 2Nh£ / Bdx 
dt dt Jo 

But from Fig. 13-7, we see that 

fd/2 /g] \ 

/ Bdx = B,x, — B, f k — x, J 

dx, 



and consequently 



v{t) = ANhB, 



dt 



(13-3) 

(13-4) 
(13-5) 



Solving Eq. (13-5) yields the spacial variation of the domain wall in 
terms of the known voltage excitation. 



x.(t) = 



4NhB, 



(13-6) 



Saturation occurs when the domain wall reaches the center of the core. 
Setting x, = d/2, we obtain 



j^' v(r) dr = 2NB.M = 2NB.A 



(13-7) 



where A = cross-section area, m 2 

B, = saturation flux density, webers/m s 
Equation (13-7) states that the time required for the complete reversal 
of the core state depends on the volt-time area of the applied voltage 
signal. Regardless of waveshape, equal area signals will result in equal 



Sec. 13-3] magnetic and dielectric devices 395 

switching times. Or expressing this result in another way, under ideal 
conditions a volt-time area greater than that given by Eq. (13-7) cannot 
be sustained across the excitation coil. 
In the special case where v(t) is a constant V, Eq. (13-6) reduces to 

x,(<) = WW. 

and we can see that the domain wall propagates at a constant velocity 
from the surface to the center. The time required for complete satura- 
tion is 

t. = ?*£* (13-8) 

In order to calculate the current flow, we turn to Maxwell's equations. 
Because of the simple geometry, they reduce to 

^ = - a E (13-9) 

dx 



dE = _dB 
dx dt 



(13-10) 



where E = electric field intensity 

a = volume conductivity of the core material 
From Eqs. (13-10), (13-4), and (13-5), 

The presence of E(t) results in an eddy-current flow in the excited portion 
of the core. This current may be considered as establishing a retarding 
field which limits the time rate of flux reversal. 

B is constant everywhere except across the domain wall. Hence the 
spatial variation of the electric field must be zero everywhere except at 
x = x,. In the center of the core, where the magnetic field has not as yet 
penetrated, E(t) will be zero (x > x,). For x < x„ the electric field 
will be given by Eq. (13-11). Since E is constant with respect to x, H 
must decrease linearly from the surface value H a (t) to the coercive force 
H c , &tx = x,. It follows that Eq. (13-9) may be rewritten as 

MLl^ = M ^='J (13-12) 

x, dt 2Nh 

where H equals Ni/l, amp-turns/m, in the mks system. 

Substituting B, dx./dt and x„ as given by Eqs. (13-5) and (13-6), into 
Eq. (13-12), we obtain the solution for the terminal current due to the 



396 



MBMOEY 



[Chap. 13 



+ /. 



(13-13) 



known voltage excitation. 

** (<)= 8™.j/ WdT 

The first term in Eq. (13-13) is the total eddy-current flow in the core; 
the second term corresponds to the current needed to overcome the 
internal coercive force. We might note that the higher the conductivity 

of the ferromagnetic material, the 
greater the core loss. If the ap- 
plied voltage is a constant, the 
current will increase linearly with 
time as shown in Fig. 13-8. 



m 


. 






f-k 


Ic 


Si-KV 2 t+I c 





ia{t) = 



ll 



8NWB. 



VH + I c 



Fig. 13-8. Terminal current response of a 
square-loop core to a constant-voltage 
excitation. 



After the core saturates, the cur- 
rent will continue to increase, but 
at a rate controlled by the induct- 
ance of the driving coil in air, its 
winding resistance, and the internal resistance of the voltage source. 

We might further note that the constant-current term in Eq. (13-13), 
due to the coercive force, may be treated as a separate bias generator in 
parallel with a postulated ideal core (zero- width hysteresis loop). This 
leads to the conclusion that the response of the core may be represented 
schematically by the models of Fig. 13-9. 



-^tffflifflT 



(a) 



o 



h N 



-ww- 

R(v,t) 



O 



(b) 



Fia. 13-9. Proposed equivalent model representations for the square-loop core. 

Response to a Known Current Excitation. As our starting point, we 
shall solve Eq. (13-12) for the location of the domain wall in terms of the 
applied magnetic field. 



*® = aSI 



[H.(t) - H c ] dr 



Substituting x,(t) back into Eq. (13-12) yields 

2Nh[H a (t) - H c ] VK 



v(t) 



V<r/ '[ff o (r)-H c ]dr 



(13-14) 



(13-15) 



Sec. 13-3] 



MAGNETIC AND DIELECTRIC DEVICES 



397 



When the input is a constant current I a , the terminal voltage and the 
saturation time are given by 



v(t) 



-4 



t. = 



i 4hWB.(I. - /«) 
alt 
<rlB,d 2 



42V(/„ - L) 



(13-16) 
(13-17) 




According to Eq. (13-16), the terminal voltage must be infinite at 
t = 0. This impossible solution is a result of the assumed ideal core 
characteristics. In practice, a finite 
time must elapse before the domain 
wall is established. Furthermore, 
the stray capacity present also 
limits the time rate of voltage 
build-up. The actual response will 
look like the dashed rather than the 
solid line of Fig. 13-10. After the 
core saturates at t = t„ the two 
terminals should appear to be a 
short circuit and the voltage should 
drop to zero. However, the large 
series impedance of the current 
source together with the small air 
inductance of the driving coil actually produces the fast exponential 
decay shown. 

From Eq. (13-17), we can see that an external excitation of /„ would 
not allow the core to saturate until an infinite time had elapsed. The 
larger the excess of current over the internal bias value, the faster the 
core will switch between the two saturation values. Here also the 
coercive force may be treated as an external bias generator as shown in 
Fig. 13-9. 

Response of an Externally Loaded Core. The voltage appearing across 
a load R connected to a secondary winding of N t turns is 

v B (t) = Nivit) 

where v(t) is the terminal voltage developed across the excitation wind- 
ing. An additional retarding field of 



t, t 

Fig. 13-10. Voltage response to a con- 
stant-current excitation of the core. 



H s (t) - 



Rl 



(13-18) 



is produced by the current flow in R. Ht(t) acts in the same direction 
as the internal coercive field, and it is simply added to H, in all the 
integral and differential equations previously derived. 



398 memoey [Chap. 13 

In the case where the applied voltage is a known function of time, the 
input current given by Eq. (13-13) would include, on the right-hand side, 
the term 

N 2 v(t) 



ir = 



R 



On the other hand, when a known current waveshape is used for 
excitation, the time-varying ff 2 must be included inside the integral 
sign. The voltage response would have to be recalculated from Max- 
well's equations. We would find that the smaller the effective load 
resistance, the larger the additional retarding field and the longer the 
time required to switch the core's state. 

From an examination of the final response to both constant-voltage 
and constant-current excitation [Eqs. (13-13) and (13-16)], we see that 
the nonsaturated core might be approximated by a nonlinear resistance 
of the form 

R < = m < 13 - 19 > 

in parallel with the current generator /„ (Fig. 13-9). K has one value 
for constant-current excitation and yet another value when a step of 
voltage is applied. 

The total resistance seen at the driving terminal is the parallel com- 
bination of the time-and-voltage-dependent core resistance R c and the 
reflected load resistance. 



Ri. 



= R (jj-j \\R e at t = 0, R c = oo, R^ = Uj-J R 



The now finite resistance limits the initial value of the voltage. The 
parallel combination is always smaller than the input resistance of the 
unloaded core, and consequently the voltage produced by a constant- 
current drive will be less than that found in Eq. (13-16). In fact, if 
RiN/Ni) 2 ^C R over the complete time and voltage range of interest, 
then the input resistance would remain essentially constant and a rec- 
tangular voltage pulse would be the result of a constant-current excita- 
tion. Since the switching time is controlled by the volt-time area [Eq. 
(13-7)], the external loading will have the adverse effect of increasing t,. 
To a good approximation, the ferromagnetic core can be considered 
to be a reasonably large resistance when it is unsaturated and a very 
small resistance (practically a short circuit) when saturated. It thus 
acts as a bivalued circuit element with, however, its switching action 
dependent on its own past history. If multiple windings are available, 
then the current flow in one can be used to control the terminal response 
at the others. For example, by alternately reflecting a high and a low 



Sec. 13-3] 



MAGNETIC AND DIELECTBIC DEVICES 



399 



resistance into the transmission path, the magnetic-circuit element can 
replace the diode bridge of Sec. 3-6 for use as a controlled gate (Fig. 
13-16). 

A Note on Units. The parameters of the ferromagnetic sample are 
usually expressed in the electromagnetic units (emu) of oersteds (H) and 
gauss (B). However, all equations used and derived are given in mks 
units. The important conversion factors are: 



Symbol 



Emu 



Mks 



Magnetizing force 

Flux 

Flux density 

Permeability 



H 
B 



Oersteds X 10V4* 
Maxwells X lO"" 8 
Gauss X 10-* 
Gauss/oersted X 4x X Vf 



Amp-turns/m 
Webers 
Webers/m* 
Henrys /m 



Example 13-1. In this example we shall compare the current response of two cores 
to an excitation of 20 volts. The first core is toroidally wound on a }£-in.-diameter 
form of 20 turns of 1-mil-thick 0.25-in.-wide Permalloy tape. The second core is a 
molded ceramic ferrite of the same size. Both cores have a 200-turn excitation 
winding. 





gauss 


Br, 

gauss 


oersteds 


« * 
jiohm-cm 




8,000 
4,500 


7,200 
3,400 


0.07 
0.18 


55 


Ferrite 


10 s 







*p is the volume resistivity, the reciprocal of the volume conductivity <r. 

Solution,, (o) The first step in the solution is the conversion of the mixed units 
given in the statement of the problem into the mks system, using the conversion fac- 
tors given above. The parameters of Permalloy are H c = 5.56 amp-turns/m, 
B, = 0.8 weber, and p = 55 X 10 -8 ohm-m. Expressing the dimensions in meters 
and substituting into the appropriate form of Eq. (13-13) results in 

t„(«) = 1.45 X WH + 1.16 X 10-* amp 

From Eq. (13-8) the saturation time of the Permalloy core is 

/, = 51.8 jisec 

At the end of the switching interval the coil current has increased to 

t.(«.) = 760 + 11.6 ma 

where the first term in the above expression is the eddy-current flow and the second 
is due to the coercive force. We might note that the low conductivity of the Permal- 
loy sample results in relatively large eddy currents. This means that additional 
power is dissipated during the switching interval above and beyond that necessary 
to overcome the coercive force. 

(6) We shall assume that the results derived for a tape-wound core apply equally 
well to the sintered ferrite core. They actually do not, but the answers obtained are 



400 memory [Chap. 13 

on the correot order of magnitude and may be used for comparison. The solution of 
Eq. (13-13), for this sample, is 

j.(0 - 14.1i + 4.1 X 10"» amp 

The ferrite core switches state in only 29 jisec. Hence the peak current flow becomes 

».«.) - 0.41 + 41 ma 

Because of the much higher resistivity of the ferrite sample, the eddy current of 410 
/ia is completely negligible. However, because of the large coercive force, the required 
driving current is still appreciable. 

13-4. Magnetic Counters. The basic magnetic counter of Fig. 13-11 
consists of two components: the core itself where the count is registered 
by the change in the magnetic state and an energy-storage element 

which automatically resets the core 

after the required count has been 
t entered. One of the more common 
v applications of this circuit is as a 

1 binary, where one output pulse is 
I produced for every two inputs. In 



r 







Fiq. 13-11. Basic magnetic-core counter. this respect the magnetic counter 

behaves similarly to the vacuum-tube 
or transistor bistable circuits of Chap. 9. The circuit of Fig. 13-11 does 
offer one major advantage in that no power is required to sustain the 
excited state. 

So that we may make use of the results of Sec. 13-3, we shall assume 
that the core is constructed of perfect square-loop material. In its 
unexcited condition, it is at —B,; we shall call this state 0. A positive 
input pulse drives it toward +B, (state 1). The differential equation 
describing the response to the applied excitation is 



iJ 



v iu = Ri + ~ I idt + v, (13-20) 



where v, is the voltage drop across the coil. 

In Sec. 13-3 we have seen that the core appears to be a very high 
resistance during the interval when it is switching from to 1 or from 
1 to 0. Since R is the small series combination of the input-source 
impedance and the winding resistance of the coil, almost the full excita- 
tion voltage will be developed across the driving coil. Equation (13-20) 
reduces to » in = v„ Hence the initial response is due to the known 
voltage excitation of the core and the current is given by Eq. (13-13). 
Until such time as the core saturates, the very small output may be 
approximated by 

v„(t) = £ idt 



Sec. 13-4] 



MAGNETIC AND DIELECTKIC DEVICES 



401 



In the special case of a constant-amplitude pulse, this reduces to 



Vo(t) 



^ (KVH* + Ij) 



(13-21) 



where K is evaluated with the aid of Eq. (13-13). 

The volt-time area of each input pulse V At is adjusted so that the 
core state will not reverse completely. The first one primes the core, 
leaving it in the partially magnetized condition corresponding to point a 
in Fig. 13-12o. After the pulse terminates, the capacitor discharges 
through the high coil resistance. But since only a small amount of 
charge was stored when the pulse was present, the demagnetization path 
follows the dashed line in Fig. 13-12a from a to b. 





B 






, 


! 








B s 




' 




■ — ■ * ■ 

b 


a 




-H c 






H c 








-B s 







H 



V- 


. 












^A(^ 










I 




t 




i 


I 






V 




1 
t 

I 
i 


\rc 


It=RC 






^-A*->- 


< 


) t x t 2 t 3 t 



(a) 



(b) 



Fig. 13-12. Response of the counter of Fig. 13-11. (a) Hysteresis curve showing the 
partial demagnetization during the discharge of the first pulse; (b) excitation pulses 
and the resultant output. 

The primed core also looks like a high resistance to the first portion 
of the second pulse. Consequently, in the interval from to h (Fig. 
13-126), the output will be given by Eq. (13-21). Once the core saturates 
at +B„ its terminal response is that of a short circuit. The capacitor 
continues charging toward V with the small time constant RC. It is 
only here that a large amount of energy may be stored in C in a short 
time interval. It follows that R must be kept as small as possible. 
Otherwise the single output pulse, which represents the count of two 
(Fig. 13-126), may never even reach V. 

At the end of the second pulse, the core is left in state 1. But as the 
capacitor starts discharging from its peak voltage V, the reverse current 
flowing in the driving coil switches the core back to state 0. Any energy 
remaining after the core resaturates at —B, is dissipated in the series 
resistance R. This portion of the cycle is identified by the double- 
segmented decay waveshape shown in Fig. 13-126. The interval from 



402 



MEMORY 



[Chap. 13 



h to t» represents the flux reversal, and the remaining portion shows the 
final discharge of C. To ensure the automatic reset after every second 
input, the energy stored in C must be greater than that dissipated in the 
core and the series resistance. Neglecting the small power loss in the 
series resistance, we obtain the necessary inequality 

YiCV* > 2B.EM 
4B,HcAl 



or 



C > 



(13-22) 



If Eq. (13-22) is not satisfied, then the discharge of C will not completely 
reset the core to state 0. The next applied pulse starts the switching 
process from a partially excited condition. It may even drive the core 
directly to state 1, leading to a false count at the output. 

We conclude that the core in the magnetic counter acts as a controlled- 
series gate, with its bivalued state dependent on its own past history. 




Reset 



9 
8 

5- 
4 
3 
—12 

-. .1 



Set 



-H 



(a) W 

Fig. 13-13. (a) An arbitrary base counter using a pulse shaper; (6) path of operation 
followed in a scale-of-10 counter. 

When unsaturated, it acts as a high series resistance which prevents the 
charging of the storage capacitor. After the core finally saturates, its 
resistance drops to zero, thus closing the charging path. 

Counting to an Arbitrary Base. If the volt-time area of the input pulse 
is carefully controlled, then this same circuit may be used to count down 
by an arbitrary factor. Assuming that the minor hysteresis loops are also 
reasonably square, each input pulse shifts the core state in discrete jumps 
along the hysteresis curve, as shown in Fig. 13-136. The final excitation 
produces saturation and resetting. For example, a scale-of-10 counter 
will result when each input pulse has a volt-time area that is slightly more 
than one-tenth of the total needed to reverse the saturation direction. 
The extra area of each pulse causes the core to saturate near the beginning 
of the tenth input and consequently permits the storage of the reset 
energy in C during the final count. 

Provided that the input signal is larger than the minimum necessary to 
switch the core state, the volt-time area of the signal developed across a 
secondary winding will depend solely on the turns ratio and the core 



Sec. 13-4] 



MAGNETIC AND DIELECTRIC DEVICES 



403 



properties [Eq. (13-17)]. Only in the unsaturated condition does the 
ferromagnetic device act as a transformer; once it saturates, the driving 
voltage will be developed across the series resistance instead of across the 
low input impedance. This means that a second core may be used to 
shape the variable input pulses to a constant area. The charge stored in 
C, in Fig. 13-13a serves to reset the shaping core after each pulse. The 
diode determines the polarity of those applied to the counting core; it 
prevents the reset pulse of the input core from registering as a negative 
count in the storage core. 

Cascading Counters. In order to achieve higher counts than are 
possible with a single core, the output of one stage may be coupled to a 
second, and so on. Each core represents the appropriate binary or 
decimal place. Because of the difficulty in reading the counts correspond- 
ing to the unsaturated states of the core (numbers between 1 and 9), 
most cascaded counters use a binary rather than a decimal base. 




Fig. 13-14. Three-stage cascaded counter. 

When several counters are directly coupled, as shown in Fig. 13-14, 
the single input must be able to supply the energy needed to reset all cores 
simultaneously. Consider the operation when the three cores are in 
state 1 (just below saturation) and the count is 1 X 2° + 1 X 2 X + 1 X 2 2 , 
reading from left to right. The eighth pulse saturates core 1, and as d 
charges, core 2 saturates, and so on, until finally C 3 is also fully charged. 
Thus we see that the volt-time area of the input pulse must be able to 
change the state of all cores and that it must also supply the energy 
that will reset all three cores. 

If more than two or three stages are cascaded, they may impose a 
severe load on the signal source. Consequently, an active regenerative 
circuit is usually included within the transmission path to decouple and to 
reshape the transmitted pulse. These circuits are discussed in Sec. 13-5. 

Shift Registers. Suppose that we wish to store the digital word 10011. 
Since it is equivalent to 

10011 = 1 X 2* + X 2 s + X 2 2 + 1 X 2 l + 1 X 2° 

this word can be converted into a train of nineteen pulses, which would 
then be injected into five cascaded binary counters. But as this requires 
an additional step, it increases the possibility that an error may be 



404 



MEMORY 



[Chap. 13 



introduced in the count. Moreover, if the word could be registered 
directly, less time would have to be allotted for storage. 

When the five digits are simultaneously available, each can be steered 
directly to the appropriate core. This form of data processing, where 
only one pulse width is needed to register the information, is called 
parallel read-in. 

On the other hand, the word 10011 may be presented in series as a 
time sequence (the serial input is pulse, pulse, blank, blank, pulse). 



Output 




Shift pulse 

Input 
pulses 



Reset 
(shift) 
pulses 



Binary 



Delay 



Binary 



Delay 























(b) 


















,. 


Input 


















, 






! i '• ! 
i i — i i ! i — i ' i i — i 




t 


Shift 














i 
i 
i 








1 i 
1 i 
i I 
i i 
















t 






















M 



















Fig. 13-15. (a) Magnetic-core shift register; (6) block diagram of any shift register; 
(c) time relationship between the input pulses (10011) and the shift pulses. 

Separating each digit there is sufficient space to insert the pulses that 
control the switching of the shift register of Fig. 13-15a. The first 
digit, i.e., the one farthest to the right in the word to be stored, is inserted 
in the normal way into core a, changing its state from to 1. The 
decoupling diode prevents the resulting negative output from switching 
the second core in the chain. 

Before the next digit appears, we apply a separate shift pulse to all 
cores (Fig. 13-15c). Those that are in the primed state will be switched 
back to 0, while those already in the zero state will be unaffected. We 
see that the counter is set by the input and reset by the shift pulse. 
The positive voltage, which will now be developed across the output 



Sec. 13-4] magnetic and dielectric devices 405 

winding of the switched core, charges the storage element C. Its dis- 
charge, through R and the primary of core 6, switches the second core 
to state 1. In order to prevent the shift pulse from interfering with the 
transferred digit, there must be a time delay in the forward transmission 
path. In the shift register of Fig. 13-15a, this delay is obtained by 
inserting a resistance in series with the driving coil of the core. 

The next digit of the word, input digit 1, again changes the state of 
core a without influencing any other core. A shift pulse will now set the 
excited cores a and b back to zero. As both capacitors discharge, the 
two digits are transferred to the right, changing the states of cores b and c 
from to 1. 

The third digit of the input word is zero. After its application the 
state of the cores is Oil — , where the dashes represent the unknown states 
due to any previous excitation. A shift pulse transfers the core states to 
001 1-. Continuing the process of first switching the state of core a and 
then transferring the stored information to the right, the full word is 
finally registered. 

Any other binary element that can be triggered on and off at two 
separate inputs, or with two different polarity pulses, can replace the 
magnetic cores in the register (Fig. 13-156). Voltage- or current-con- 
trolled negative-resistance devices are ideally suited for use as memory 
elements in shift registers. If bistable multivibrators are used, a simple 
RC network can be used to delay the transfer until after the shift pulse 
terminates. 

For read-out, five shift pulses will sequentially feed the stored digits 
to the output terminal. This process is identical with the normal shifting 
which transfers the input forward from core to core. In fact, as a new 
word is entered (starting with a shift rather than a digit pulse), the old 
word is automatically shifted out on the right. 

One obvious disadvantage is that the stored word is erased as it is 
fed out for examination or use. However, if the shift register is formed 
into a closed loop, then the information fed out on the right can be auto- 
matically re-stored in the left-hand core. After the appropriate number 
of shift pulses are applied, the register again contains the original word. 

A convenient means of closing the self-storage path is by way of a 
controlled gate such as shown in Fig. 13-16. This circuit makes use of the 
bivalued resistance properties of the core to interrupt or close the feed- 
back path. During storage the gate is kept open (the gating core is in its 
active region) and the register functions normally. When reading out 
information temporarily stored, the gate is also left open. But if we 
desire more permanence, then the gate must be closed (core-saturated) 
during the intervals that the output pulses are produced. It is implicitly 
assumed that the amplitude of the transferred pulse will not change the 



406 



MEMOBT 



[Chap. 13 



state of the magnetic gate and that the reflected control signal will not 
register as a storage pulse. 

Ring Counter. A closed-loop shift register can also be used as an 
arbitrary base counter. One and only one core (or binary) is set to 
state 1. Then as each succeeding input pulse is applied to the shift 
windings, the single stored digit is transferred sequentially around the 
loop. For example, of the 10 cores necessary in a decimal counter, 
initially only the zero core is excited. The first pulse transfers the single 
digit to core one, and so on. The tenth input switches the ninth core 
from to 1 and resets the zero core to 1. This same voltage pulse may 
also be coupled to a second ring counter, which is used to indicate the next 
highest decimal place. 



Control signal 




Input 



Sp Output 



Fig. 13-16. Closed-loop shift register — a switched core used as a controlled gate. 

It follows that the particular count registered is indicated by the 
number of the single excited core. If the volt-time area of the transfer 
pulse is carefully controlled, then the core can be switched from —B. 
to the verge of +B,. In these circumstances the difference in the termi- 
nal resistance seen across an auxiliary winding readily identifies the 
location of the digit. In fact, if the current flow through the read-out 
winding is always less than J c (so that it cannot influence the magnetiza- 
tion), then the individual core will act as a simple controlled gate, 
inserting a high resistance during the count and a low one at all other 
times. The gating process can, of course, be reversed by transferring 
a rather than a 1 around the loop. 

13-5. Core -transistor Counters and Registers. The number of stages 
which can be directly coupled in a counter or shift register is limited by 
the power capabilities of the pulse source. In the counter, the input 
must be able to switch all cores simultaneously. In the register, the 
shift pulse not only resets all cores but must also supply the energy 
that transfers the stored information to the right. And since the switch- 



Sec. 13-5] magnetic and dielectric devices 407 

ing time depends on the excess of magnetizing current over the coercive 
force [Eq. (13-17)1, the problem is further complicated when the counting 
speed becomes critical. For example, in order to switch a typical core 
in a 250-kc register, the l-jisec shift pulse applied must have a peak power 
of almost 1 watt. The 330-ma pulse current will develop an average 
of 3 volts — and a peak which may be many times as large — across the 
shift coil. Under these conditions we can expect a single transistor 
amplifier to switch, at the most, three or four stages. 

One obvious means of improving the response is to use an amplifier 
to decouple the individual cores. An even better way is by inserting a 
regenerative circuit between two cores ; the output of the first core can be 
made to trigger the generation of a fast-rising high-power pulse for 
propagation in the forward direction. 



JL * 



Input 




Output 



Bias 



Bias 



Fro. 13-17. Cascaded counter using a transistor blocking oscillator for regeneration 
and a core for storage. The input amplifier stage is also shown. 

Figure 13-17 shows a counter which is a happy combination of a core 
and a transistor. This circuit employs a single core both as a memory 
(winding Ni) and as a transformer for the blocking oscillator. Besides 
the economy in using one rather than two cores, the sharing improves the 
response because the oscillator pulse also, resets the counter. In the 
normal state the core is at and the transistor is cut off by the external 
bias. The first pulse brings the core to the verge of saturation, and the 
second one saturates it and charges C x . With the transformer connec- 
tions shown, these input pulses act in a direction that keeps the transistor 
off. 

After the termination of the second input, the voltage across N\ reverses 
as Ci starts discharging. This pulse, coupled to N 2 , brings the transistor 
into its active region. And as the regenerative circuit goes through one 
complete cycle, the extra energy supplied by the collector current rapidly 
returns the core to state 0. 

In the counter of Fig. 13-17, the functions of regeneration and storage 



408 memory [Chap. 13 

are separately handled by the transistor and the core. The active 
element is operative only during the reset interval. At all other times, 
the passive core controls the circuit behavior. The capacitor only 
triggers the regenerative circuit; it does not have to supply the energy 
for reswitching. Hence the restrictions on C are much less severe, and 
this circuit will recover much faster than those depending only on the 
stored charge for reset. 

From which point in the circuit shall we take the output? If the 
collector is chosen, all the transformed input pulses will also be coupled 
to the next stage. However, as these are of the opposite polarity to the 
pulse produced by the blocking oscillator, a series diode will prevent 
them from registering falsely at the next core. This diode may be 
eliminated if the output is taken at the junction of the collector resistor 
and the transformer winding, as shown in Fig. 13-17. Here the cutoff 
transistor effectively replaces the decoupling diode. 

As an alternative, the output may be taken from a fourth winding 
on the core. Since the switching during recovery is between the two 
saturation levels, this pulse would have the stabilized volt-time area 
which is especially necessary when counting to a base other than 2. 
By properly choosing the turns ratio, the output volt-time area can be 
adjusted to any desired percentage of the reset pulse. This avoids the 
use of a separate core for shaping the counting pulses that are applied 
to the next stage. 

If, instead, the fourth winding is used as an input for the shift pulses, 
then the train of cores and transistors will function as a shift register. 
When the core is in the zero state, the low impedance offered to the shift 
input will not permit the development of a pulse large enough to bring 
the transistor out of cutoff. After the information pulse primes the core, 
the opposite acting shift pulse can trigger the regenerative circuit. Dur- 
ing reset, the previously stored information is transferred one position 
to the right. 

13-6. Magnetic Memory Arrays. Figure 13-18 illustrates the arrange- 
ment of a compact large-capacity memory unit. We shall assume that 
each of the 25 small ferrite cores exhibits the ideal square-loop properties 
of Fig. 13-19. Before information is stored by changing its state, the 
individual core is saturated at —B,. Three insulated leads thread the 
core: one passes through each row, a second through each column, and a 
third threads the complete array. 

Suppose that we wish to switch the state of the single core 34, located 
at x = 3 and y = 4. Positive current pulses of amplitude 7„ are simul- 
taneously applied to row 3 and column 4. The total field in the core 
located at the intersection of the excited row and column (due to 27 a ) 
must cause switching. However, as all other excited cores should remain 



Sec. 13-6] 



MAGNETIC AND DIELECTBIC DEVICES 



409 




Fig. 13-18. A 5 X 5 magnetic-core coincident memory, 
in their original state, the pulse amplitude must satisfy 

y 2 Ic </.</„ (13-23) 

Only core 34 is excited past H c (to point 2H„ in Fig. 13-19). All the 
other cores in the row and column have a field strength of only H a ; 
after the termination of the pulse they relax back to the original state. 
Thus the injection of the two pulses carrying the address of the core 
switches its state and stores the indi- 
vidual digit. These pulses may be 
directed properly with the aid of 
auxiliary steering cores or diode gates. 

In order to read out the stored 
digit, the memory unit must be inter- 
rogated. To do so, negative current 
pulses ( — la) are injected into the 
appropriate row and column. If the 
core situated at the intersection is in 
an excited state, it will switch back 
to 0. The voltage developed during 
the transition appears at the termi- 
nals of the read-out winding which 
threads all cores (Fig. 13-18). Suppose that the particular core interro- 
gated is in the zero state. In this case the low impedance presented 



-#, 



B s 



H a 



2H a 



H c 



B. 



H 



Fig. 13-19. Hysteresis loop showing 
excitation of switched and non- 
switched cores of the memory array 
of Fig. 13-18. 



410 



MEMORY 



[Chap. 13 



would permit the development of only a very small output, which could 
not possibly be mistaken for a digit. Since only a single core is ques- 
tioned at a time, there will not be any ambiguity as to the location of the 
output information. 

We should note that the process of read-out is destructive ; it erases the 
stored information. External circuitry has been devised which auto- 
matically resets the excited core. The presence of the output digit 
triggers the generation of a positive current pulse immediately after the 
termination of the negative interrogation signals. Nondestructive stor- 
age and read-out depending on special core geometries have also been 
devised, but a discussion of these is beyond the scope of this text. 

Switching time in the array of Fig. 13-18 is on the order of 1 /*sec. 
In order to reduce the access time, smaller cores must be used. But 
doing so increases the problem of wiring the large arrays needed. Instead 
of using individual cores, many high-speed memory units depend on 
switching small regions of ferrite or magnetic film. The conductors are 
first printed on an insulated sheet. At each "intersection" a small 
amount of ferrite or metallic film is deposited. Holes punched through 

the "core" region provide for the 
insertion of the read-out winding. 
With the small amount of magnetic 
materials used, switching times of 
20 to 50 m/isec have been obtained. 
A further advantage is that this 
type of memory lends itself to the 
economical production techniques 
of automation. 

13-7. Core -transistor Multivi- 
brator. In Sec. 13-3 we have seen 
that the saturation time of a 
square-loop core, excited by a con- 
stant voltage, is very clearly de- 
fined. The abrupt change in the 
driving-point impedance, which 
marks the end of the flux build-up, 
permits the core to be used as the 
timing element in a monostable or an astable circuit. 

Figure 13-20 shows a single-core multivibrator employing resistive 
coupling to complete the regenerative loop. In this circuit, the winding 
direction is such that when Ti is saturated, the current through 2Vi 
drives the core from — B, to +£,. Current through the second winding 
Nz, due to saturation of T 2 , resets the core state to — B,. As in all 
astable multivibrators, the transistors furnish the greater-than-unity 



rvws 
iMMJ 



i — nfflS^ 



nmp — i 

iV 2 • 




Fig. 13-20. Core-transistor multivibrator 
used as a d-c to d-c or a d-c to a-c con- 
verter. 



Sec. 13-7] 



MAGNETIC AND DIELECTRIC DEVICES 



411 



loop gain needed to switch between the two quasi-stable states. They 
become active only during the small switching interval corresponding to 
the saturation of the core. At all other times the individual transistor 
is either saturated or cut off. 

We shall now detail the operation over one complete cycle, assuming, 
as the starting point, that Ti is off, just turning on, and T* is on, just 
turning off. This is equivalent to saying that the core is on the verge of 
saturating at — JS,. As long as the flux is still changing, the full supply 
voltage will appear across the high driving-point impedance of N% and 
the collector voltage of T* will be approximately zero. Hence no current 
can flow into the base of T t ; it is cut off. 




En ]_ 




-VA- 



± 



if I VW-1 ° 



fa) 



2E bb 



<b) 



Fig. 13-21. Models holding during the change of flux density from — B, to B. in the 
core of Fig. 13-20. (o) Saturated transistor T\\ (6) back-biasing of cutoff transistor TV 

Once the core saturates, there is a marked decrease in the coil's driving- 
point impedance. Under these circumstances an appreciable voltage 
is developed across the collector-to-emitter saturation resistance of the 
series transistor 5P». Current can now flow into the base of the off 
transistor, and as T\ turns on, the current through the winding iVi starts 
the flux reversing. En is connected across this winding, as shown in the 
model of Fig. 13-21a. The transformer action reverses the voltage 
across JV 2 . The resultant large increase in the base current of Ti rapidly 
drives this transistor into saturation. At the same time, the drop in the 
collector voltage of T\, coupled through the biasing and speed-up capaci- 
tor Ca, will turn T 2 off (Fig. 13-216). 

The time that Ti remains saturated is controlled by the flux build-up 
in the core. If the small demagnetizing base current is neglected, then 
the core switching time, as given by Eq. (13-8), is 



h = 



2NiAB. 
Ehj, 



412 



MEMORY 



[Chap. 13 



After h sec the core saturates at +B, and the circuit switches to its other 
quasi-stable state. 

The charge stored in the speed-up capacitor aids in maintaining the 
base of the off transistor below zero, thus stabilizing the circuit's opera- 
tion. From Fig. 13-2 la we can see that as long as Ti is saturated, 
Ci is in series with the transformer winding JV 2 and Em,. When JVi = iV 2 , 
d will charge toward 2E bb . Meanwhile C 2 is discharging through R 2 
as shown in Fig. 13-216. The conditions which must be satisfied in 
order to sustain the base of the off transistor below zero for the full half 
cycle are 

4r 2 > h 4n > h 
where t 2 = R 2 C 2 and n = R x d. 

Suppose that C 2 is too small; will its discharge before the end of the 
half period cause the premature switching of this circuit? Since the base 




Fig. 13-22. Transistor collector and base waveshapes — circuit of Fig. 13-20. 

of the off transistor T 2 is returned to the collector of the saturated one, 
T h only a very small base current can possibly flow into TV Its collector 
current, flowing in winding iV 2 , cannot institute regenerative switching. 
It only opposes the flux build-up due to the much larger current in 
Ni. We thus conclude that the core still controls the timing. 

If the two timing coils are identical, this circuit will generate the 
square wave shown in Fig. 13-22. The frequency of oscillation will be 
given by 

' - k - wjb, (13 - 24) 

when the transistor switching time is neglected. It is interesting to 
note that the square-wave frequency is linearly dependent on the supply 
voltage. 

The output is usually obtained from a third winding; hence the trans- 



Sec. 13-7] 



MAGNETIC AND DIELECTBIC DEVICES 



413 



Output 




Fid. 13-23. A transformer-coupled 
core-transistor multivibrator. 



former action not only affords isolation but also permits the amplitude 
to be scaled up or down. 

Figure 13-23 shows another core-timed multivibrator, one which 
utilizes transformer rather than resistive coupling between the active 
elements. In this circuit, the volt- 
ages developed across the auxiliary 
windings Ni and Nt sustain the one 
transistor in cutoff and the other in 
saturation during the flux build-up 
interval. Consider the conditions 
when the saturated transistor Ti con- 
nects Ebb across winding N* With 
the directions shown, winding iVi sup- 
plies the saturation base current of 
T\. The voltage appearing across 
Nt back-biases Ti. These voltages 
are easily calculated once the turns ratios are known. 

The half cycle terminates when the saturation of the core permits the 
transistors to reenter their active regions. Any decrease in voltage 
across the driving coil is coupled by the transformer into the base of T 2 , 
turning it on. Since the circuit is regenerative, switching continues until 
Ti saturates and Ti cuts off. The flux then starts reversing, and the 
cycle repeats. 

These same circuits are widely used as d-c to a-c or as d-c to d-c con- 
verters. The primary peak-to-peak voltage of 22?«, (measured from 
collector to collector) is first multiplied by the primary to secondary 
winding ratio n. If direct current is desired, the square wave, which is 
present at the output, must be rectified. Thus a 10-volt battery may be 
used to develop a 100-volt (or even higher) a-c or d-c output. Of course, 
the primary source impedance is multiplied by n 2 as it is reflected into the 
output circuit while the voltage is only increased by the factor n. Thus 
the PR losses increase faster than the voltage. This limits the per- 
missible step-up ratio, especially when large amounts of power must be 
supplied to the load. In any case, power transistors having a very low 
saturation resistance should be chosen. 

When this circuit is used as a d-c to a-c converter, the frequency of 
operation is usually the critical factor. It will even determine the 
transformer design, i.e., the turns ratio, cross-section area, and type of 
magnetic material employed. Since the frequency is inversely pro- 
portional to the product of the transformer parameters, the greater the 
turns ratio, the smaller the volume of iron needed. However, increasing 
the number of turns also increases the winding resistance and the losses; 
■therefore some compromise must be selected. 



414 memory [Chap. 13 

On the other hand, when operating as a d-c to d-c converter, the 
higher the frequency of oscillation, the easier it is to filter any a-c com- 
ponents appearing in the output. As a further advantage, both the 
amount of transformer iron and turns needed decrease with increasing 
frequency. But if the frequency of operation becomes excessively high, 
the stray circuit capacity will adversely affect the waveshape. Since 
the transistor will remain active over a larger portion of the cycle, the 
effective source impedance also increases, resulting in poorer regulation 
with respect to load changes. 

For example, a typical converter used in an automobile will operate 
from the 12- volt battery and will supply a maximum of 1 amp to the 
load at a nominal 120 volts and 60 cps. In this application the total 
primary resistance may be estimated at 0.1 ohm, 0.05 ohm due to the 
saturation of the power transistor and the other 0.05 ohm contributed 
by the winding and battery resistance. Since the transformer's turns 
ratio is 10: 1, 10 ohms will be reflected into the output. At full current 
flow, the output drops to 110 volts. The primary voltage falls by the 
same percentage and causes a frequency shift down to 55 cps [Eq. (13-24)]. 
While this converter might be used to operate many small appliances, the 
frequency change under load would prevent its use for phonographs or 
other small constant-speed motors. 

13-8. Properties of Ferroelectric Materials. Dielectrics are those 
substances where all charged particles are relatively tightly bound to the 

atomic nucleus or the molecular 
region. In the presence of an ex- 
ternal field, these charges cannot 
move freely through the material 
to the surface as do the relatively 
loosely bound electrons in conduc- 
tors. Instead, there will be a slight 
separation in the individual micro- 
scopic region; the positive charge 

Fig. 13-24. Individual dipole. moVeS in the direction of the E 

field, and the negative charge 
moves in the opposite direction. Hence the local effect on the field 
pattern is that of ah elementary dipole, i.e., an associated positive and 
negative charge (q) separated by some small distance d (Fig. 13-24). 
In certain dielectrics these dipoles exist even before the external field 
is applied. But since they are randomly oriented, the material does not 
exhibit a net field of its own. It follows that the driving-point character- 
istics of the dielectric may be determined from the motion and alignment 
of the elementary dipoles under the influence of the applied field. 

Figure 13-24 shows an individual dipole from which the potential field 




Sec. 13-8] magnetic and dielectric devices 415 

at any point can be calculated. If the point is relatively far from the 
charge, the potential is given by 

gdcos^ yolts (13 . 25) 

where to, the permittivity of free space, is equal to 8.85 X 10 -12 farad/m 
in the mks system. The dipole moment is defined as 

p = qd 

and p is directed from the negative to the positive charge. This vector 
is called the polarization of the dipole. In any small volume, where 
there are n dipoles, the total polarization is the vector sum of the indi- 
vidual dipole moments. When they are all aligned, P = np, and when 
they are in random array, P = 0. 

In order to express the terminal characteristics in simple terms, we 
define the displacement vector 

D = *>E + P , (13-26) 

The relationship between P and E is, in general, quite complex; for 
example, the polarization in a single direction in a crystal may depend 
on all three components of the applied field. However, for the ideal 
dielectric, we can assume the linear relationship 

P = to kE (13-27) 

which leads to 

D = «<,€ r E 

where e r = 1 + fc is called the relative permittivity (or dielectric constant) 
of the medium. In the ferroelectric materials with which we shall be 
concerned, fc 2> 1, and the electric displacement is almost exactly equal 
to the polarization. 

One class of dielectrics can be defined as those materials where the 
polarization is produced solely by the external field. For these to be 
useful in capacitors, the relationship between P and E should be the 
linear one expressed in Eq. (13-27). If it is not, then the dielectric 
constant, and hence the capacity, will vary with the applied voltage. 

Certain organic waxes exhibit a quasi-permanent residual polarization 
when they are solidified in the presence of an electric field. These are 
called electrets, and they are the closest electrostatic analogs to the 
permanent magnet currently existing. 

In the third and, for our purposes, the most important class of dielec- 
trics, the polarization is a nonlinear function of E. Here the material 
is characterized by spontaneous polarization, saturation, and hysteresis 
(Fig. 13-25). So as to recognize the duality with the magnetic materials, 



416 



MEMORY 



[Chap. 13 



these dielectrics are unfortunately called ferroelectric. They contain 

no iron, nor is the duality complete. 

Ferroelectricity was first observed in rochelle salts, NaK(C4H 4 6 )- 

4H 2 0, by J. Valasek, in 1921. The 
usable temperature range of this 
substance is only 40°C, and because 
of its complicated crystal structure, 
it develops ferroelectric properties 
in one direction only. Barium 
titanate, BaTi0 3 , is much more 
useful since it is ferroelectric be- 
low its Curie point of 120°C. The 
crystal has a simple perovskite 
structure. Many other substances, 
such as triglycine sulfate and guan- 
idine aluminum sulfate hexahy- 
drate, also exhibit ferroelectric 
properties. Some of these other 




Fig. 13-25. Typical hysteresis loop of 
ferroelectric materials. 



materials have squarer hysteresis loops, higher Curie temperatures, or 
better long-term storage properties than barium titanate. 

The switching behavior of the ferroelectric crystal has been extensively 
investigated with the aid of polarized light and optical examination. In 
the unsaturated sample, small domains are found where the net dipole 
moment is in one of the permissible directions. Studies with a thin plate 





(a) (b) 

Fig. 13-26. Domain switching, (a) 180° domains showing the spike formation; (b) 
90° domains showing the wedge formation. The shaded regions have been switched 
by the external field. 

cut from a single crystal of barium titanate indicate that adjacent 
domains may have polarization directions either 180 or 90° apart. In 
the 180° domains, the application of a field E > E c causes the formation 
of spikes of new domains in the preferred direction (Fig. 13-26a). These 
then extend across the whole crystal. The 90° domains are reversed by 
the formation of wedges skewed to the cathode wall (Fig. 13-266), which 
grow and spread sideways, eventually covering the entire region. 
The number of both the 90 and 180° wedges formed, and hence the 



Sec. 13-9] 



MAGNETIC AND DIELECTRIC DEVICES 



417 



switching time, is a function of the excess of field strength over the 
internal coercive force. 

13-9. Ferroelectric Terminal Characteristics. The practical form of 
the ferroelectric device is a thin wafer of barium titanate, or some other 
ferroelectric material, cut from a single crystal. The two electrodes 
must be carefully deposited on the flat surfaces because any air gap that 
exists will decrease the effective dielectric constant and seriously degrade 
the squareness of the observed hysteresis loop. 






^ 



^ 



v d 





D. 


Q. 








-V c 


Vc 


V 




-E c 


E c 




~e" 




-D, 


-Q. 


(b) 





(a) 

Fig. 13-27. (a) Schematic representation of a ferroelectric device; (6) ideal square- 
loop characteristics. Both the microscopic (E,D) and macroscopic (Q„F.) coordi- 
nates are shown. 



In a parallel-plate capacitor of thickness d and area A, 



D = 



Q 



and 



E = 



V a 



(13-28) 



where V a is the terminal voltage. Furthermore, the charge flow is simply 



«-/: 



idt 



For a known current excitation, the time required to reverse the satura- 
tion direction of a perfect square-loop dielectric (Fig. 13-27) is given by 



Q« 



= 2Q, = T' i dt = 2D.A 



(13-29) 



Equation (13-29) states that a fixed current-time area (charge) is 
required to switch the dielectric. Or, expressing the relationship of 
Eq. (13-29) in a more useful manner, the ferroelectric device acts as a 
series charge regulator. It permits only the fixed charge packet Q m 
to flow during the switching intervals. This equation should be com- 
pared with the equivalent relationship previously derived for the ferro- 
magnetic core [Eq. (13-7)]. Under the special condition where the 



418 memoky [Chap. 13 

applied current is constant (t = /«), the time required to switch from 
— D, to +D., or vice versa, is 

. _ 2D.A 

During this interval the terminal voltage increases linearly with time, 
starting from the coercive value V c . 

After switching, the ideal ferroelectric device can no longer transfer 
any charge. Consequently, it seems reasonable to approximate the 
saturated response by an open circuit. 



/-constant 



V— constant 




t. 
(a) 



(b) 



Fio. 13-28. Terminal response of ideal ferroelectric device, 
excitation; (6) constant-voltage excitation. 



(o) Constant-current 



When a voltage excitation is applied, the switching time is found 
to be inversely proportional to the excess over the coercive force. For a 
constant voltage V a , the relationship is 



t. = 



K 



V e 



(13-30) 



where if is a constant that depends on the geometry and properties of 
the ferroelectric medium. Equation (13-30) is exactly analogous to 
Eq. (13-17). 

To a very good approximation the voltage and current response of 
the ferroelectric device during switching will look like the current and 
voltage response of the ferromagnetic core, respectively (Fig. 13-28). 
Where our concept of the core was a time-varying resistor decreasing 
from infinity as the core approached saturation, our model of the square- 
loop ferroelectric device is a resistance that increases from zero with time 
and current. It appears in series with a bucking voltage equal to the 
internal coercive force V c . Sometimes the model of the ferroelectric 
element is further simplified, by neglecting the energy loss during switch- 
ing, and it becomes a bivalued capacitor. In fact, when the termination 
of the excitation leaves the ferroelectric element unsaturated, its incre- 



Sec. 13-10] magnetic and dielectric devices 419 

mental response is most like that of a large capacitor. Since the relative 
dielectric constant drops from several hundred down to unity as the 
material saturates, the capacity is reduced to that of the two deposited 
electrodes in free space. 

The widely used ferroelectric material barium titanate has a coercive 
field strength of 1,500 volts/cm. Only by using extremely thin slabs 
(0.033 to 0.167 mm) can the coercive voltage V c be kept between 5 and 
25 volts. Over the transition interval the average resistance of the ferro- 
electric element ranges between 100 and 1,000 ohms. Because of the 
large remnant polarization, 22 X 10 -8 coulomb/cm 2 , a large charge 
packet is transferred during switching, even from quite small elements. 

13-10. The Ferroelectric Counter. The principle of charge monitor- 
ing is employed in the counter of Fig. 13-29. As the input square-loop 



"in 



?HQr 



Output 




Input 



Fig. 13-29. Ferroelectric counter. 



ferroelectric element (FEi) is switched from negative to positive satura- 
tion by the leading portion of the input pulse, a charge packet of 2Q, 
flows through Z> x into the storage element C. This increases the output 
voltage by the definitive increment 



AF„ = 



2Q. 

C 



(13-31) 



The negative portion of the input pulse resets FEi to negative saturation 
with £> 2 establishing the current path. During this part of the cycle 
the back-biased diode Z>i prevents the discharge of the previously stored 
count. 

Note that the input pulses must be properly shaped. They must 
have positive peaks greater than V c + V , as well as a duration greater 
than the switching interval. The negative peak can be smaller, but 
its magnitude must still exceed V c . Otherwise the ferroelectric device 
will not switch and reswitch between the two saturation limits and will 
not meter the full charge packet to the storage capacitor. 

During the final count the change in the output voltage brings the 
regenerative circuit into conduction. For example, the final change of 
A V,, volts may be used to trigger a monostable multivibrator or a blocking 



420 



MEMORY 



[Chap. 13 



oscillator which propagates a single count in the forward direction into 
the next stage. This same pulse will turn on the shunting transistor, 
discharge C, and reset the counter. 

The switching action of the counter can be greatly improved by the 
simple modification shown in Fig. 13-30, i.e., the replacement of C by a 
second ferroelectric element. Each input pulse will be shaped into a 
charge packet of 2Qi by FEi. This charge also flows into FE 2 and starts 
it switching from — Q a2 toward + Q,2- If the second element is larger, 
n charge packets (or input pulses) will be needed to saturate FE 2 . 
Until such time as it does saturate, the ferroelectric element appears 



7?E, 

-HQr- 
input 



-M- 



Kd 2 



>R, ±, 



Regenerative 
stage 



D 3 

-M- 



Output 



CDFE 2 



1 



_J 



Fig. 13-30. Improved ferroelectric counter. 

to be a low impedance and only a very limited voltage can be developed 
across it. 

If FE 2 becomes saturated during the nth input, then it may be replaced 
by a very high impedance for the remaining portion of the pulse. The full 
excitation will appear across the output storage element, and instead of 
depending on identifying the small charge of AV„ volts, the large voltage 
jump presents a definitive trigger to the regenerative stage. With the 
modification of Fig. 13-30 this counter can accurately register many more 
counts than possible with the simpler circuit. Furthermore, since the 
count is registered by a change in the state of the element and not by the 
storage of charge, it will be less affected by leakage resistance. Counters 
having rates as low as one per day and counting ratios as high as 30 : 1 or 
40:1 have been successfully operated. 

In order to reset the storage element, a negative pulse can be applied 
through a diode from some point in the regenerative stage. However, 
unless resistance is inserted in series with D 2 , the two input diodes would 
act as a short circuit, preventing the resetting of FE 2 . Current will also 
flow into FEi during the reset interval, but this only aids its recovery 
toward negative saturation. 

Example 13-2. Another interesting application of the pulse area shaping by the 
ferroelectric and ferromagnetic devices is the tachometer shown in Fig. 13-31. The 
sharp input pulses obtained from the distributor in a six-cylinder 2-cycle engine are 
assumed shaped to switch and reset the charge-metering ferroelectric element. Their 



Sec. 13-11] magnetic and dielectric devices 421 

amplitude and duration will, of course, be erratic. When switching from negative to 
positive polarization, the input element delivers a charge packet of 20 X 10~* cou- 
lomb. We wish the highly damped d-c voltmeter reading of 5 volts to correspond to 
a speed of 5,000 rpm. 




D-c 
Input * C5fc R< (J voltmeter 



Fig. 13-31. A ferroelectric tachometer (Example 13-2). 

Three input pulses are generated on each revolution. Hence the minimum spacing 
between the individual inputs, which occurs at the maximum velocity, is 

„, 60 sec . 

= 4 msec 



" 3 X 5,000 

So that the storage capacitor can discharge completely between the adjacent pulses, 
the output time constant must be much less than T m . Assuming that the charging 
time is insignificant and that T > 4t, the average voltage read on the d-c meter is 
given by 



V d .. = ± JJ *V e-'/' dt = AV 



T 



where &V is the peak output voltage due to the metered charge packet. Note that 
the output RC circuit provides a d-c voltage inversely proportional to the spacing 
between the pulses and directly proportional to the motor rpm. 
At 5,000 rpm, 

AF t = ^=r RC - 20 volt-msec 

Thus R = 1,000 ohms. Setting T = 500 yusec yields C = 0.5 rf. Since AV = 40 volts 
and V c ■» 10 volts, for proper switching the peak positive input must be greater than 
50 volts. The diode allows satisfactory resetting with negative peaks greater than 
10 volts. 

13-11. Coincident Memory Arrays. The ferroelectric element is some- 
what more difficult to use in a memory matrix than the ferromagnetic 
core. Since only two electrodes are available, these must be used not 
only for storage and interrogation but also for read-out. It is not pos- 
sible to place a third electrode at the memory position, as it was to thread 
the complete core array with a third winding. Consequently various 
ingenious circuits have been devised to obtain the output digit. % 

Consider the memory plate of Fig. 13-32, where the x and y direction 
leads are printed at right angles on the opposite faces of the ferroelectric 
plate. The intersections are the memory positions, and they may be 
treated as individual ferroelectric elements. In order to obtain the 
coincident switching, voltage inputs must be applied to the two leads. 



422 



MEMORY 



[Chap. 13 



To store a digit in a particular location, that memory element is switched 
from negative to positive polarization by applying a positive pulse to the 
j/i row and a negative one to the Xj column. Each pulse must have an 
amplitude falling within the limits 

YiV c <\v a \< v. 

and a duration sufficient to cause switching. 



V a - 



4=U 



TT 



TT 



ru 



3-4-C 



11 
1 



Storage pulses 



Read-out pulses 



*1 *2 x 3 * 

Memory matrix 



Fig. 13-32. A ferroelectric memory matrix. 

It is quite simple to reset the memory element from 1 to 0. If it 
contains a stored digit, pulses of the polarity opposite to those used 
for storage will sum at the intersection and reverse the polarization. 
All other positions remain unaffected. But how may the digit be 
registered after interrogation? The switching process results in the 
transfer of a charge pocket which must somehow be monitored. In one 
method, shown in Fig. 13-33a, a resistor is connected in the ground return 
of either the x or y pulse generator. The current flow during storage 
will cause a negative voltage to appear across the resistor. During 
reset, the opposite polarity signal is developed. The two output pulses 
are easily distinguished, and if desired, a shunting diode may be incorpo- 
rated to eliminate the unwanted negative pulse. 

Figure 13-336 illustrates a somewhat superior method of producing 
an output after interrogation. All the x leads (or y leads) are threaded 
through a single read-out core. The current flow during the transfer 
of the metered charge packet induces a pulse into the output winding. 
One p*olarity output is produced on storage, and the opposite on reset; 
the diode shown selects the correct signal. 

One major disadvantage of the ferroelectric memory matrix is that the 
state of the particular element is affected by single-voltage pulses of less 
than the coercive force. In barium titanate it was discovered that 
multiple queries in any one row caused the partial depolarization of all 



MAGNETIC AND DIELECTRIC DEVICES 



423 



memory positions subjected to the half-voltage pulse. After several 
hundred interrogations the polarization may even be reduced to half 
of the saturation value. It has been found, however, that crystals of 
other materials, such as triglycine sulfate, appear to have more nearly 





(a) 
Fig. 13-33. Two methods of monitoring the read-out pulse. 

ideal characteristics. As these sustain full polarization under repeated 
questioning, they will no doubt be used in practical memory banks. 

PROBLEMS 

13-1. A square-cross-section Permalloy core measuring J£ in. on a side has an effec- 
tive magnetic path length of 6 in. It is excited by a 10-volt peak-to-peak square 
wave applied across a 20-turn winding. 

(o) What is the minimum-duration square wave needed to drive the core between 
its two saturation limits? 

(6) Plot the spatial penetration of the field and the current build-up as a function 
of time when the duration of the half period is twice as long as that found in part o. 

In the course of the calculations assume a source impedance of 10 ohms and wind- 
ing resistance of 1 ohm where necessary to keep the solution within bounds. 

13-2. Plot the spatial variation and current build-up for the core of Prob. 13-1 for 
the following excitations. After the core saturates, the current is limited by the 
100-ohm series resistance; this resistance may be neglected during the interval that 
the core remains unsaturated. 

(a) A 50-volt peak-to-peak 30-^sec-period sine wave. 

(6) A 50-volt peak-to-peak 30-jisec-duration sawtooth. 

(c) A 50-volt peak-to-peak 30-/usec-period triangular wave. 

13-3. Consider the two models shown in Fig. 13-34 which have been suggested to 
approximate the core response during the switching interval. In order to see the 





Fio. 13-34 



424 memory [Chap. 13 

advantages and disadvantages of these representations, we shall compare the exact 
and approximate response to a known voltage excitation. 

(a) Calculate the model constants (R, k, and h) for the Permalloy core of Example 
13-1. The current at t = t, of both models should agree with that given in the text 
for constant-voltage excitation. 

(b) Plot on the same graph the exact and the model current response to a voltage 
ramp which increases at the rate of 5 volts/jisec. It drives the core from negative to 
positive saturation. 

Which model gives the best representation? Is there any modification of the model 
which will result in a closer correlation? Explain. 

13-4. Plot the incremental driving-point resistance as a function of time for the 
two cores described in Example 13-1. Will this resistance change with excitation? 
In your answer consider the response to a voltage ramp, which increases at a rate of 
10 volts /Vsec. 

13-5. Find an equivalent resistance for each of the cores of Example 13-1 that will 
yield equal power dissipation over the switching interval. Treat the following two 
cases and compare the results. 

(a) Constant-voltage excitation (20 volts). 

(6) Constant-current excitation (/ = 10/ e ). 

13-6. Repeat Prob. 13-1 when the core is excited by a square wave of current hav- 
ing a peak-to-peak value five times as large as the coercive force. In this case plot 
the terminal voltage response and the location of the domain boundary with respect 
to time. 

13-7. Calculate and sketch the voltage response of the core of Prob. 13-1 to the 
following current excitations. The core is initially at negative saturation, and each 
signal reverses the flux direction. 

(a) A current ramp which increases at the rate of 20 ma//isec. 

(h) A single cycle of a sine wave having a peak-to-peak amplitude of 50 ma and a 
duration of 5 jusec. 

13-8. (a) Solve for the expression of the voltage response to a unit step of current 
when the core is loaded across a secondary winding N 2 with a resistor R. 

(6) Plot the response of the Permalloy core of Example 13-1 to a 20-ma unit step. 
The external loading consists of 2,000 ohms across a 10-turn winding. 

(c) Repeat part 6 when the load resistance is decreased to 100 ohms. 

13-9. In the counter of Fig. 13-11 the ferrite core is wound with 20 turns; it has an 
effective radius of 3.0 cm and a square cross section of 1.0 cm on a side (its parame- 
ters are those given in the text). C = 0.001 ^f, and R = 500 ohms. 

(a) Specify the range of the volt-time areas of the input pulses for which this 
counter will count down by the ratio 4:1. What is the minimum spacing between 
these pulses? 

(b) Repeat for a counting ratio of 8:1. 

(c) Sketch and label the output when the core is excited by a train of 20-volt 
5-jisec-wide pulses. These are widely spaced compared with the recovery time of 
the core. 

13-10. A ferrite core having a radius of 0.5 in., d = 0.2 in., and h = 0.3 in. is used 
as the memory element in the simple counter of Fig. 13-11. One output pulse should 
be produced for every three of the 25-volt 10-jusec input pulses. During the two 
priming inputs, the output must remain below 5 volts. 

(o) Specify the minimum value of C that will properly reset the core on the termi- 
nation of the count. 

(b) Using twice the minimum value of C, specify the range of core turns for which 
this counter will work. (The RC time constant is 1 usee.) 



MAGNETIC AND DIELECTRIC DEVICES 



425 



(c) Sketch and label the output voltage when C is twice the minimum value and 
when JV lies in the center of the acceptable range. 

13-11. Two identical ferrite cores having the dimensions given in Prob. 13-10 are 
used in the scale-of-10 counter of Fig. 13-35. Complete the design by specifying C. 




Fig. 13-35 

needed to reset the input core. Calculate the minimum input volt-time area and the 
pulse spacing necessary for proper counting. Discuss the circuit modification neces- 
sary to ensure the automatic reset of the counting core. 

13-12. (a) Draw the circuit of a transistor multivibrator shift register, indicating 
where and how to inject the input and the shift pulses. Show three stages. Indicate 
the time delay between the stages. 

(b) Repeat part a when a tunnel diode is used as the memory element. Pay 
particular attention to the coupling network and to the method of injecting the input 
and the shift pulses. 

(c) Neon bulbs which fire at 100 volts, which have a conduction drop of 75 volts, 
and which require 0.5 ma to sustain conduction might be used to indicate the count 
in a core decade ring counter. Explain how they would be incorporated into the 

circuit. 

(d) Exercise your ingenuity to see if it is possible to ensure the reset of the core ring 
counter, regardless of the stored digit, by the injection of a single pulse. If neces- 
sary, the basic circuit may be modified. 

13-13. Figure 13-36 shows a magnetic core and gate incorporating automatic 
reset. Each coil consists of 5 turns wound on the single ferrite core (radius of 3 cm 
and with a cross section measuring 1 cm by 1 
cm). 

(o) Sketch the output voltage when 1-msec 
current pulses, having an amplitude equal to 
0.4/ c , are simultaneously applied to two 
inputs; to three inputs. 

(6) If C » 0.001 A what is the minimum 
pulse width that will ensure the automatic 
reset of the core? The loss in R may be 
neglected in the calculation. 

(c) How might this circuit be converted 
into an or gate? 

13-14. The transistor-core multivibrator 
of Fig. 13-20 is used as a d-c to d-c con- 
verter producing 150 volts from the 20-volt source. It operates at approximately 
1,000 cps, which simplifies the filtering at the output of the full-wave bridge rectifier. 
Each of the 10-turn primary windings has 0.1 ohm resistance. The saturation 
resistance of the transistor is 0.2 ohm, and at the boundary between the active and 
saturation regions, p may be taken as 5. Grain-oriented silicon steel, havmg the 
following parameters, is used for the square cross-section core: B, = 2 webers, B, = 
12 amp-turn/m, r = 40 X 10"» ohm-m, and I = 0.3 m. 




SJVk 



Fio. 13-36 



426 



MEMORY 



[Chap. 13 



0.05 mm 



(a) Calculate the maximum value of R and the minimum value of C necessary to 
sustain one transistor off and the other one on over the full half cycle. 

(6) What cross-section area is needed for the core to operate at 1,000 cps? 

(c) Calculate the total power dissipated in the converter when the external load 
current is 150 ma. How much power is dissipated under no-load conditions? 

(d) Sketch and label the current and voltage waveshapes at the base and collector, 
under loaded conditions. 

13-16. A silicon-steel core, having the characteristics given in Prob. 13-14, is used 
in the multivibrator of Fig. 13-23. For this circuit the cross section measures 2 cm 
on a side and the magnetic path is 20 cm long. The primary consists of four windings 
of 5, 15, 10, and 5 turns, which are connected to the base and collector of the first 
transistor and to the collector and base of the Second transistor, respectively. The 
secondary is a single 50-turn winding. 

Sketch and label the collector and base current and the output voltage waveshapes' 
when Ebb = 20 volts. Assume a base-to-emitter resistance of 0.02 ohm for your 
calculations. 

13-16. (a) Plot the potential as a function of 9 far from an individual dipole. 
Assume that r is constant and equal to 50d. 

(b) On the same graph as in part o plot the constant potential distance r as a func- 
tion of 9. The absolute value of the po- 
tential should be equal to the maximum 
value found in part a. 

13-17. The incremental capacity of 
0.2 mm the wedge-shaped ferroelectric element 
shown in Fig. 13-37 is a function of the 
applied direct current. Calculate this 
capacity variation, neglecting any fringe 
effects. The dielectric used has t, = 2,000 
Fio. 13-37 and E c = 1,500 volts/cm, and its cross- 

section area is 0.25 cm. (Hint: First show 
that the capacity of a large-area parallel-plate capacitor is C = e e,A/d.) How 
will the resonant frequency of a tank circuit, using this element, vary with the d-c 
voltage? 

13-18. (a) Prove that the charge packet flowing during the switching interval in a 
ferroelectric element is equal to the charge transferred in a parallel-plate capacitor of 
the same dimensions as it charges from — V c to V c . 

(6) The feiroelectric element FE t described in Prob. 13-19 is charged from a 
20-volt source through a 5,000-ohm resistor. After saturating, « r drops to unity. 
Plot the current and voltage response. Calculate the approximate time-varying 
resistor 

B(t) **At+B 

that will result in equal charge flow over the switching interval. Furthermore, the 
terminal voltage at t = t, should be equal to V c - 

13-19. The counter of Fig. 13-38 is excited by two input sources, each connected 
to the single storage capacitor through a separate charge-metering element. FEi 
consists of a 0.03-mm-thick slab having an effective surface area of 0.02 cm 2 , while 
FE S is twice as thick and has only half the electrode area. The ferroelectric material 
has a coercive field strength of 2,000 volts/cm and a remnant polarization of 8 X 10 _ « 
coulomb /cm 8 . We may further assume that the constant in the switching equation 
(13-30) is K = 10 volt-Msec. 




MAGNETIC AND DIELECTRIC DEVICES 



427 



+20 



v 



-20 



*. 

"20 




EE? ~i I 



Fig. 13-38 




Trigger 



Sketch the output voltage, giving all significant values, if the Bhaped 20-volt-peak 
20-/isec-wide pulses are applied at the two inputs at 

«i at t = 0, 10, 20, 30, . . . msec 
e% at t - 5, 15, 25, 35, . . . msec 

13-20. Figure 13-39 shows a monostable multivibrator where the duration of the 
quasi-stable state is controlled by the 
switching time of a ferroelectric element. 
In order to simplify the calculations it will 
be represented by a 0.005-/xf capacitor which 
decreases by a factor of 1,000 after saturat- 
ing at ± 5 volts. Sketch and label the volt- 
age at the collector of Ti and at the collec- 
tor and base of T 2 after a pulse is injected 
into the base of Ti. Does this type of 
timing offer any advantage over that ob- 
tained from a simple capacitor? 

13-21. Show that the circuit of Fig. 
13-39 can be modified so that it can make 
use of a ferromagnetic core to control the 
timing. If the core of Prob. 13-10 is employed, sketch and label the important 
waveshapes. 

BIBLIOGRAPHY 

Bates, L. F.: "Modern Magnetism," 3d ed., Cambridge University Press, New York, 

1951. 
Chen, Kan, and A. J. Schiewe: A Single Transistor Magnetic Coupled Oscillator, 

Trans. AIEE, pt. I, Communs. and Electronics, vol. 75, pp. 396-399, September, 

1956. 
Chen, T. C, and A. Papoulis: Domain Theory in Core Switching, Proc. Symposium 

on Role of Solid State Phenomena, Polytech. Inst. Brooklyn, April, 1957; also in 

Proc. IRE, vol. 46, no. 5, pp. 839-849, 1958. 
Collins, H. W.: Magnetic Amplifier Control of Switching Transistors, Trans. AIEE, 

pt. I, Communs. and Electronics, vol. 75, pp. 585-589, November, 1956. 
Dekker, A. J.: "Electrical Engineering Materials," Prentice-Hall, Inc., Englewood 

Cliffs, N.J., 1959. 
Katz, H. W.: "Solid State Magnetic and Dielectric Devices," John Wiley & Sons, 

Inc., New York, 1959. 



Fig. 13-39 



428 memobt [Chap. 13 

Little, C. A.: Dynamic Behavior of Domain Walls BaTiOs, Phys. Rev., vol. 98, no 4 

pp. 978-984, 1955. 
Menyuk, N.: Magnetic Materials for Digital Computer Components, /. Appl. Phys., 

vol. 26, no. 6, pp. 692-697, 1955. 
Meyerhoff, A. J., and R. M. Tillman: A High-speed Two-winding Transistor-mag- 
netic-core Oscillator, IRE Trans, on Circuit Theory, vol. CT-4, no. 3, pp 228- 

236, 1957. 
Rajchman, J. A.: A Myriabit Core Matrix Memory, Proc. IRE, vol 41 no 10 pp 

1407-1421, 1953. 
: A Survey of Magnetic and Other Solid-state Devices for the Manipulation of 

Information, IRE Trans, on Circuit Theory, vol. CT-4, no. 3, pp. 210-225, 

September, 1957. 
Royer, G. H. : A Switching Transistor D-C to A-C Converter Having an Output Fre- 
quency Proportional to the D-C Input Voltage, Trans. AIEE, pt. I, Communs. 

and Electronics, vol. 74, pp. 322-326, July, 1955. 
Rozner, R., and P. Pengelly: Transistors and Cores in Counting Circuits, Electronic 

Eng., vol. 31, pp. 272-274, May, 1959. 
Sands, E. A.: An Analysis of Magnetic Shift Register Operation, Proc. IRE, vol 41 

no. 8, pp. 993-999, 1953. 
Storm, H. F.: "Magnetic Amplifiers," John Wiley & Sons, Inc., New York, 1955. 
Von Hippel, A.: "Dielectric Materials and Applications," John Wiley & Sons Inc 

New York, 1954. 
Wolfe, R. M.: Counting Circuits Employing Ferroelectric Devices, IRE Trans, on 

Circuit Theory, vol. CT-4, no. 3, pp. 226-228, 1957. 



PART 5 

OSCILLATIONS 



CHAPTER 14 

ALMOST SINUSOIDAL OSCILLATIONS— 
THE LINEAR APPROXIMATION 



The earlier chapters of the text showed how the generation of many- 
periodic waveshapes (rectangular pulses and linear sweeps) was predi- 
cated on driving the active elements of the circuit far into their saturation 
and/or cutoff regions. The timing function depended upon the non- 
active regions of the system, with the active zone simply supplying the 
energy necessary for switching between the two exponential timing 
regions. 

As we start discussing the generation of almost sinusoidal signals, it 
should be pointed out that the nonlinearity of the system will now play 
a subsidiary role. The basic timing is due to a frequency-determining 
network, with the amplifier simply setting the necessary conditions for 
oscillation. To avoid distorting the output signal, the degree of non- 
linearity must be small, and consequently most of the subsequent dis- 
cussion can concern itself with the linear approximation of the sinusoidal 
oscillator. 

14-1. Basic Feedback Oscillators. Figure 14-1 presents the basic 
configuration of a feedback oscillator as a block diagram. Even though 
each of the three essentials shown 
is not completely isolated, it is con- 
venient to think of them as sep- 
arate entities. It is especially 
helpful to segregate the complete 
nonlinearity so that it can be given 
individual attention. The remain- 
ing elements in the circuit may then 
be treated by the standard methods 
of linear analysis. 

The behavior of the circuit of Fig. 14-1 is defined by two equations, 
one for the forward transmission and the other giving the feedback 
voltage. 

e» = («i + e f )AL e t = ffez 
431 



ei+ef 


A 


-^ «2 


Ue) 






r~ 




e 3 


1 


_« 








p(u>) 






* 











Fio. 14-1. Basic-feedback-oscillator con- 
figuration showing the ideal amplifier A, 
the amplitude limiter Lie), and the fre- 
quency-determining network /S(u). 



432 oscillations [Chap. 14 

From these equations, we find that the over-all closed-loop gain is 

(? = £?= AL (14.!) 

ei 1 — /SAL v ' 

For self-sustained oscillations, there must be an output without any 
external excitation. This becomes possible only when the denominator 
of Eq. (14-1) vanishes. It follows that the frequency and amplitude 
of oscillation must satisfy 

1 - p(fi>)AL(e) = (14-2) 

where 0AL is simply the open-loop transmission (i.e., the switch in Fig. 
14-1 is open). Equation (14-2) implies that the feedback must be 
regenerative. 

Limiting is often accomplished by driving one or more of the amplifier's 
active elements into their saturation or cutoff regions for a portion 
of the cycle. Alternatively, an amplitude-controlled resistor or other 
passive nonlinear element may be included as part of the amplifier or 
in the frequency-determining network. If a gross nonlinearity is per- 
mitted, the limiter will distort the signal and the output will be far from 
sinusoidal. 

For small signals no limiting will occur and L(e) will take on its maxi- 
mum value of unity. It follows that if Afi > 1 in the small-signal region, 
the amplitude will build up until the limiter stabilizes the system at an 
output level that satisfies Eq. (14-2). Thus, in the active region, the 
threshold loop transmission that permits self-sustained oscillation is 

PA, = 1 

This equation is called the Barkhausen criterion for oscillation. 

Unity loop gain at a single frequency is a necessary but not a sufficient 
condition for self-sustained oscillations. If the network characteristics 
are such that the net phase shift is zero at several frequencies, then the 
criterion for oscillation can be arrived' at only from an examination of the 
complete amplitude and phase portrait of the system. Nyquist stated 
that the polar plot of Ap for -co < oi < a> must encircle the point 
1 + jO for oscillation. However, for many of the simpler configurations, 
the Barkhausen condition will be both necessary and sufficient. Below 
we shall consider circuits that are known to oscillate and that can be 
analyzed on the basis of their having unity loop gain. 

To ensure our objective, the generation of a sinusoidal signal, the 
frequency should be primarily determined by a network whose character- 
istics can be rigidly controlled. This network must contain at least two 
energy-storage elements so that the system response [Eq. (14-2)] will 
have the necessary pair of complex conjugate roots which give the 



Sec. 14-1] 



ALMOST SINUSOIDAL OSCILLATIONS 



433 



natural frequency of oscillation. For simplicity, A may be assumed 
to be a constant, with its frequency dependence lumped together with 
that of 0. 

Since A in Eq. (14-2) now represents the gain of an ideal amplifier, 
it will be a pure number having a sign corresponding to the stages of 
amplification (minus for an odd number and plus for an even number). 
P represents the transmission characteristics of the frequency-selective 
network and will therefore be a function of w. To satisfy Eq. (14-2) 
the imaginary part of f)A must vanish. Since the complete frequency 
variation is included in the /3 network, the imaginary part depends only 
on j8. Consequently the frequency at which 



Imp = 



(14-3) 



will be the approximate frequency of oscillation. 

At the oscillation frequency, the remaining portion of Eq. (14-2), 
that is, Re 03 A), must be identically equal to unity. The value of gain 
that will just ensure the sinusoidal oscillations will be given by 



or 



Atfim) 
A t 



j8(«o) 



(14-4) 






X 

T 1 I 




This value of threshold gain is measured with the amplifier output loaded 
by the input impedance of the net- 
work. The open-circuit gain must, 
of course, be somewhat larger. 

Equations (14-3) and (14-4), taken 
together, state that the over-all loop | 
gain must be unity and the over-all g* 
phase shift must be zero at the fre- 1 
quency of oscillation. 

Consider the circuit of Fig. 14-2, 
where we shall assume that the 
amplifier is adjusted so that it will 

just sustain the oscillations. The frequency-determining network is 
characterized by 

R 



«z 



Fig. 14-2. Example of a feedback oscil- 
lator. 



m = % = 



pCR + 1 



pili R 

n ^~ pG^ pCR + 1 



3 + pCR + 



1 



pCR 



Substituting ja for p results in 
= 



1 



3 + i(cu/o) — Wo/«) 



(14-5) 



(14-6) 



434 



OSCILLATIONS 



[Chap. 14 



where a> = l/CR. The imaginary portion of the denominator of Eq. 
(14-6) vanishes at 

1 



O)o = 



CB 



At this frequency £ = J^, and from Eq. (14-4) we find that the gain 
necessary to sustain oscillation is A t = 3. To achieve this positive gain, 
the amplifier must contain an even number of stages. 

Because they determine the nature of the transient response, it is 
interesting to examine the location of the roots of 1 — 0A, and hence 
the poles of the complete feedback system as a function of A. The 
roots of the circuit of Fig. 14-2 are found by substituting Eq. (14-5) 
into 1 - 0A = 0. This leads to 



P 2 + 



3- A 
CR 



P + 



(n)'-° 



(14-7) 




Equation (14-7) has two roots which traverse the path shown in Fig. 14-3. 

As A increases from to 1, they first 
coalesce at —l/CR on the negative 
real axis and then separate along the 
semicircular paths shown, finally 
reaching ±j/CR on the imaginary 
axis for A = 3. This condition cor- 
responds to the threshold of oscilla- 
tion, with the location of the com- 
plex conjugate roots giving the 
frequency. 

When the system has two distinct 
complex conjugate roots, i.e., for A 

between the limits 1 < A < 5, the transient response will be given by 

e(t) = Ke°" cos (out + <j>) 
And the two roots are located at 

Pl,2 = OC ± jbll 

For small values of a, the natural frequency is 



Fig. 14-3. Migration of the roots of 
Eq. (14-7) as a function of A. The 
arrows indicate increasing gain. 



where a>o = y/a 2 + ui 2 . 
almost o>o- 

If A is less than 3, then the real part of the root will be negative 
and any oscillatory response will damp out. The degree of damping 
depends on the distance from the poles to the imaginary axis and hence 
on the value of A. 

Once the poles move into the right half plane, the real part forces an 
increasing exponential build-up. But the voltage is bounded by the 



Sec. 14-2] almost sinusoidal oscillations 435 

limiter; the output eventually stabilizes at a peak amplitude that makes 
the average gain over the cycle equal to 3. At the peaks of the sine 
wave, the limiter reduces the loop gain, driving the system poles from the 
right- into the left-half plane. The amplitude decays, the circuit reenters 
the active region, and the roots move back into the right half plane. 

If the poles are initially close to the imaginary axis, then the build-up 
will be quite slow and small changes in the root location will have but 
little influence over any single cycle of the sine wave. When they are 
located far into the right half plane, the exponential build-up makes itself 
felt during each cycle; the waveform will include a greater degree of 
distortion. Even with a small gain margin over the threshold value, 
the location of the roots will be such that the oscillator will generate a 
signal of slightly different frequency than that calculated on the basis of 
the borderline behavior. 

The oscillator will be self-starting only if initially the poles lie in the 
right half plane. It follows that the necessary gain must be somewhat 
in excess at A,. Any infinitesimal disturbance will cause an amplitude 
build-up until the limiter stabilizes the output. The threshold gain 
sustains existing oscillations, but does not provide a margin for build-up. 
Excess gain will also stabilize the circuit in that any slight reduction in A 
will not stop the oscillations when A > A t but will cause them to damp 
out when the circuit operates marginally. 

If the amplifier gain ever becomes large enough to drive the roots onto 
the positive real axis, that is, A > 5, the system response will no longer 
be oscillatory. The exponential build-up will lead to a relaxation 
phenomenon similar to that found for the astable multivibrator. 

14-2. Characteristics of Some BC and LR Frequency -determining 
Networks. The self-starting oscillator drives itself into limiting and in 
the process distorts the sinusoidal output signal. Harmonics, introduced 
by the limiter, may be treated as additional signals injected within the 
feedback loop. Each harmonic term will also be affected by the feedback 
present ; it will be acted upon by the factor 

where /3„ is the feedback factor evaluated at the particular harmonic under 
investigation. 

Since we wish to maintain an almost sinusoidal output, we should 
ensure that the amplitude of the harmonics produced in the limiter will 
be small compared with the fundamental. Each distortion term is further 
modified by H n . With a properly designed network, the feedback will 
change from positive to negative as the loop excitation frequency increases 
from wo to 2wo. The magnitude of H„ will become less than unity at the 



436 oscillations [Chap. 14 

second-harmonic and even smaller at the higher-harmonic frequencies. 
Thus the nature of the existing feedback is changed and is used to reduce 
the amount of distortion. 

The more selective the network, the greater the degree of limiting 
which can be tolerated without excessively distorting the final sinusoidal 
output. Consequently |.ffs| serves as a measure of the network's quality 
for oscillator application. For consistency it will always be evaluated 
under threshold conditions, instead of using the slightly larger value 
of the actual small-signal gain. The best network, all other things being 
equal, is the one having the lowest value of H 2 . 

The output signal, which is of the form 

e 3 = E l cos (coot + <t>i) + Y H„E n cos (nu4 +• <(,„) (14-9) 

n = 2 

is transmitted back to the input through the £ network. Each term is 
also multiplied by /3„, resulting in an effective driving signal of 

00 

ei = faEi cos (wot + 00 + Y p n H„E„ cos (nuot + <*>„) 

re-2 

In order to have a true basis of comparison as to the relative amplitudes 
of the distortion components at the input and at the output, it is advisable 
to use the normalized input harmonic factor 

K. = ^ (14-10) 

Equation (14-10) accounts for the relative attenuation of the fundamental 
component which would be ignored if n H„ were chosen instead as the 
figure of merit. When \K n \ = \H n \, the relative harmonic contents of the 
input and output are equal. In any specific circuit the best signal, i.e., 
the most nearly sinusoidal, will appear at the output for \K„\ > \H n \ and 
at the input for \K n \ < \H n \. 

Phase-shift Network. One network commonly used in the so-called 
"phase-shift oscillator," which we must therefore name "the phase- 
shift network," appears in Fig. 14-4a. Either the three series branches 
(Zi) will be energy-storage elements L or C and the shunt branches 
(Zi) resistors, or vice versa. In general, all series branches and all 
parallel branches are composed of equal elements; however, this is not 
an essential condition. Satisfactory oscillators have been constructed 
with widely unbalanced sections and with networks of more than three 
sections. 



Sec. 14-2] almost sinusoidal oscillations 437 

From Fig. 14-4o, the network transfer characteristics can be found to 



be 



= ^ = 



(14-11) 



e 3 (Z 1 /2 2 ) 8 + 5(Zx/Z 2 ) 2 + 6Zi/Z 2 + 1 
Since either Z\ or Z%, but not both, is an energy-storage element, the 





Zx 










Zx 










Zi 










1 












:l 


13 






Z 


2 








z 


2 








z 2 


^ 




<b) (c) 

Pig. 14-4. (a) Basic "phase-shift network"; (6 and c) two specific examples. 

odd-power terms in the denominator of Eq. (14-11) contribute the 
imaginary part of P(w). For it to vanish 



(!) 



" + 6|!-0 



(14-12) 



In the specific case of the RC network of Fig. 14-46, Z\ = 1/juC and 
Zt = R; substituting these values into Eq. (14-12) yields 



&>o = 



1 



Vqcr 



(14-13) 



As the elements in the network change, so will the form of the equation 
defining coo- 

By either substituting the value of &> from Eq. (14-13) intoEq. (14-11) 
or by making the simpler substitution of 



(!)' = -< 



from Eq. (14-12) into Eq. (14-11) and by noting that the odd powers of 
Z1/Z2 have vanished, the threshold value of /? becomes 



0(«o) = Pi 



V* 



29 



438 



OSCILLATIONS 



[Chap. 14 



Thus the amplifier must have a minimum gain of —29 for sustained 

oscillations. With this particular 
type of network, the required gain 
is independent of the elements com- 
prising Zi and Zi. Any other RC 
or LR combination will give the 
same value of gain but a different 
operating frequency and network 
input impedance. Since the neces- 
sary gain is negative, a single or an 
odd number of stages are required. 
Figure 14-5 illustrates one possible 
circuit. 
,| for the RC network of Fig. 14-4&, ft, 




A single-tube phase-shift 



Fig. 14-5. 
oscillator. 

In order to evaluate \H 2 \ and \K\ 
may be expressed as 



1 



1 - 30/n 2 - j(Q V6/n)(l - 1/n 2 ) 



where n represents the order of the harmonic. Taking A t = —29 and 
substituting into Eqs. (14-8) and (14-10) yields 



\H*\ 



0.368 
1.25 



For this network, the waveshape containing the smallest amount of har- 
monics appears at the amplifier output. If the positions of R and C are 
interchanged, this will no longer be true; the best signal would now appear 
at the input to the base amplifier. 

Variation of the oscillator frequency over the widest frequency range 
requires the simultaneous adjustment of three similar elements. In the 
network of Fig. 14-46, any changes in jB will also change the driving- 
point impedance. This may load the base amplifier to a point where 
oscillations can no longer be sustained. If C is varied, then we face the 
problem of tracking three independent variable capacitors. Because of 
these difficulties, the phase-shift network is generally used for a fixed-fre- 
quency oscillator. Here only minor calibration adjustments are needed, 
and they may be made by means of a single small trimmer capacitor or 
padding resistor. 

The Wien Bridge. The simple RC network used as an example in 
Fig. 14-2 exhibits extremely poor selectivity. Any second-harmonic 
components introduced are not only not reduced, but are actually 
increased in amplitude. In this network \H t \ - 2.24 and \K 2 \ = 2. 

But all is not yet lost. We can convert the simple network into a null 
balance bridge (Fig. 14-6) and exchange necessary loop gain for selectiv- 



Sec. 14-2] almost sinusoidal oscillations 439 

ity. In the bridge 

e/ = et — ez 

but et/d = of the simple network of Fig. 14-2 [Eq. (14-6)]. The 
resistor combination Ri and Ri is ad- 
justed to satisfy the condition 

ez = Ri = 1 _ 1 
ei R t + Ri 3 8 

Thus the over-all 0' of the bridge of 
Fig. 14-6 may be expressed as 



18'- 



- £> - 



ei 



P 



where 




(H) 



(14-14) 




1 



3 + j(<»/(>>o — ojo/oj) 



Fio. 14-6. A Wien-bridge frequency- 
selective network. 

The amplitude and phase response 

of /S' are plotted in Fig. 14-21. At the resonant frequency of the system 
(coo = 1/CR), &' reduces to 

P = T 



and the threshold gain A t is equal to 8. The term 1/8 may be considered 
as the degree of bridge unbalance; when 8 = », the bridge is perfectly 
balanced at a> 0> and when 8 = 3, the circuit reduces to the simple network 
of Fig. 14-2. If follows that the greater the degree of balance, the larger 
the necessary loop gain to sustain oscillations. At perfect balance, an 
infinite gain is required; with any physical amplifier, the circuit cannot 
oscillate. 

The improvement in selectivity may best be seen by substituting /3' 
given by Eq. (14-14) and A, = 8 into the expression for H n [Eq. (14-8)]. 
The result is 

3 (14-15) 



H' = 



8(1 - 3j3„) 



where /3„ is the transmission factor of the RC network at the harmonic 
frequency. Taking the ratio of H' n for the bridge to H„ found for the 
completely unbalanced case (8 = 3) shows the dependence of the selec- 
tivity on 8. 

H± _ 3/[8(l - 3ft,)] _ 3 

H n 1/(1 - 3ft.) 8 

From Eq. (14-16) we conclude that the percentage of second harmonics 



(14-16) 



440 



OSCILLATIONS 



[Chap. 14 



present at the output is reduced in proportion to the increase in required 
amplifier gain. With a two-stage amplifier having a gain of 300, the value 
of \H' t \ is on ly 0.0224, a hundredfold reduction from the unbalanced case. 
At the output of the bridge, i.e., at e s in Fig. 14-6, the improvement in 
selectivity is not as impressive. K'„ may be expressed by 

# , = 3 - 6(1 - 3ft.) 
#"" 8(1 -3A.) 



K> 



(14-17) 



As 5 gets large, the magnitude of Eq. (14-17) approaches unity as a limit. 
For other than very large gains, \K' n \ will always be slightly above unity. 
To incorporate the Wien bridge into an oscillator, we need two iso- 
lated input terminals, usually the grid and cathode of the input tube as 
shown in Fig. 14-7. Transformer coupling may also be used to convert 




Fig. 14-7. A Wien-bridge oscillator. 

the double-ended network output to a single-ended amplifier input. 
With a transistor amplifier, a transformer would almost always be used 
to avoid loading the bridge by the low-impedance input. 

In this circuit, we see that positive feedback for regeneration is supplied 
through the RC branches to the grid of the input tube. The pure 
resistive path introduces a negative-feedback voltage into the cathode. 
The combination of both terms controls the operation of the circuit. At 
wo the positive feedback predominates, and at the harmonics the net 
negative feedback reduces the distortion components. 

The basic circuit of Fig. 14-7 is used in many commercial wide-range 
audio oscillators of from 10 cps to 200 kc or even higher. To adjust 
the frequency, usually both resistors are changed by steps and C is 
changed smoothly as a fine control. We can also do the reverse — switch 
C in steps and vary R continuously. Both arrangements perform 
satisfactorily. 

Other Null Networks. Almost any three-terminal null network may be 
used as the frequency-determining pair of branches in a bridge. The 
other two arms are composed of resistors adjusted for the proper degree 
of unbalance. 



Sec. 14-2] almost sinusoidal oscillations 441 

All arguments employed with respect to the Wien bridge will also 
apply to these other networks, provided that care is taken as to the direc- 
tion of the unbalance. For some networks the greatest reduction in 
harmonic content comes when 5 is positive, as in the Wien bridge, and for 
others S must be negative. Depending on the network and the sign of 




I — VW — t — WV — I 
■1 — II — r II 1—9 




(b) 
Fig. 14-8. Two null networks, (a) Bridged-T network; (6) twin-T network. 

unbalance at &jo, in some cases the resistive branches must be used for the 
positive feedback, and in others the reactive branches so serve. 

Figure 14-8 shows two networks which give zero transmission at their 
null frequency. The feedback factors and further defining relationships 
are tabulated in Table 14-1. A plot of the amplitude and phase response 



Table 14-1 



Network 



Transfer function 



Denning terms 



Bridged-T. 



-■ 



1-j 



Qo ta* — «o* 



"' LC 
r 
2rC 



Twin-T. 



-■ 



1 -j2 



k -+- 1 Utifg 

\/k OJ 2 — Id 



1 



2RR 2 C* 



R o C 

k = = 2 — 

2Rt C 2 



appears in Fig. 14-21. At the oscillation frequency there cannot be any 
feedback through the reactive elements and the necessary positive feed- 
back must be supplied through the resistive path. The basic circuit 
configuration needed to satisfy this condition is shown in Fig. 14-9. 
Here the value of /3' may be expressed as 



P 



Ri -J- R2 



- P 




T 



442 oscillations [Chap. 14 

and at the null point, the required threshold gain is 

A t = 1 + Jl 

As in the Wien bridge, when A, is large, the harmonic content of the out- 
put will be small. 
To adjust the frequency of these two networks, we must vary three 

elements simultaneously. The dif- 
ficulty that this entails makes 
these networks best suited for fixed- 
frequency operation. For the same 
degree of unbalance, the bridged-T 
circuit will usually give the cleanest 
waveshape. However, since it is 
more difficult and expensive to ob- 
tain a high-quality inductance than 
Fig. 14-9 Oscillator connection when us- - t ig tQ find extremely good resist ors 
ing a null network for frequency deter- . ■ • m i 

mination. and capacitors, the twin-1 network 

might be preferred. 

14-3. Transistor Feedback Oscillators. The relatively low base input 
impedance of the transistor often prevents its direct substitution for 
the high-input-impedance vacuum tube in the various oscillator circuits 
previously discussed. Even after scaling down the resistors and increas- 
ing the capacity in the frequency-determining network, the heavy load- 
ing makes it more difficult to establish the necessary unity loop gain. 
For example, if the phase-shift network of Fig. 14-4 were connected 
directly between the collector and base of a suitably biased transistor, 
the output of this network would be heavily loaded by the input imped- 
ance of the transistor. Even though it is possible to sustain oscillations, 
the frequency would be extremely dependent on the transistor parameters 
—a relatively unsatisfactory situation. In order to solve this problem, 
many transistor oscillators resort to some form of impedance matching 
between the output of the feedback network and the input voltage- 
amplifier stage. 

Figure 14-10 shows a transistor phase-shift oscillator which employs 
an emitter follower as the impedance transforming stage. It presents a 
high input impedance across the output of the frequency-determining 
network and a low output impedance for coupling to the input of the 
amplifier stage. By using this additional transistor, it now becomes 
possible to make R « (1 + /3) R& and still keep the input impedance of the 
phase-shift network large compared with the collector load R 3 of Ti. 
The emitter follower has effectively isolated the elements controlling the 
frequency from the remainder of the circuit. 



Sec. 14-3] 



ALMOST SINUSOIDAL OSCILLATIONS 



443 



Since at the frequency of oscillation the attenuation from e 2 to e x is 
— M9. the base amplifier must have a minimum loaded gain (A = e*/ei) 
of —29 to sustain the oscillation. Generally, the open-circuit (unloaded) 
voltage gain of T t would have to be from 20 to 50 per cent higher. 




Fig. 14-10. A transistor phase-shift oscillator: T x is the voltage amplifier, and IT, an 
emitter follower used for decoupling. The phase-shift network consists of three 
identical RC sections where- R a ^Rb = R- 




Fig. 14-11. A transistor Wien-bridge oscillator using a composite transistor input 
stage. 

A second method of increasing the effective input impedance is illus- 
trated by the Wien-bridge oscillator of Fig. 14-11. A composite transis- 
tor, which was discussed in Sec. 7-4, is used as the input stage of the base 
amplifier. The two transistors Tj. and T t , taken together, have a base-to- 
emitter input impedance that is approximately (1 + /3) times as large 
as that of a single transistor. Provided that reasonable values of 
resistors are chosen for the bridge arms, the input characteristics of 
the composite transistor will not adversely affect the selectivity of the 
bridge. In order to decouple the bridge from the high-impedance 



444 



OSCILLATIONS 



[Chap. 14 



collector as well as to have a low-impedance point from which to take the 
output, we could also insert an emitter follower after TV All the results 
of the discussion of the Wien bridge in Sec. 14-2 are directly applicable 
to this circuit. 

Current-controlled Oscillations. If the transistor is used in its natural 
mode, with the appropriate form of feedback, then the oscillations pro- 
duced may be said to be current- 
controlled. The basic circuit con- 
figuration is shown in Fig. 14-12. 
Except that i replaces e, it is 
identical with the voltage-con- 
trolled block diagram of Fig. 14-1. 
To avoid confusion between the 
voltage and current-feedback fac- 
tors and with the forward-current- 
amplification factor of the tran- 
sistor, we shall designate the 



*l h+if 


S 


N 












S 






'/ 


. 








' 























r 



Fig. 14-12. Basic current-controlled oscil- 
lator: current amplifier A c , limiter L(i), 
and frequency-determining network T(a). 



current feedback by T(u). This term is defined by 

if = T(u)i 2 

and as before, the complete frequency dependence of the system is 
assigned to T(a>). 




Fig. 14-13. An example of a current-controlled oscillator. 

The conditions for self-sustained oscillations (ii = 0) may be written 
by analogy with Eq. (14-2). 

1 - T(f*)AMQ = ( 14 " 18 > 

Equation (14-18) states that the net loop current gain must be unity at 
the frequency and amplitude of oscillations. 

Figure 14-13 illustrates a circuit that is the current-controlled equiv- 
alent of the simple oscillator of Fig. 14-2. By assuming that the load 



Sec. 14-4] 



ALMOST SINUSOIDAL OSCILLATIONS 



445 



impedance of the final stage is large compared with the input impedance 
of the network, the full output current will flow into T(oi). If this 
assumption is not justified, the current will divide proportionally between 
Rl and the network and Rt will influence the oscillation frequency. 
For the ideal case, 

6 + J \oi/ wo — uo/w) 

where wo = 1/RC. [Compare this equation with Eq. (14-6).] The fre- 
quency of oscillation is a>o, and the threshold current gain must be 3. 




Fig. 14-14. An oscillator using a tuned circuit as a selective network. 

The only conditions which we must impose on the various RC and 
RL networks, which may be used for feedback, are that T(w) must give 
a clearly defined oscillation frequency and good harmonic rejection. 
Also, to avoid coloring the network response, the associated current 
amplifier should approach the ideal, i.e., a zero input impedance, a high 
output impedance, and constant gain over the frequency range of interest. 

14-4. Tuned-circuit Oscillators. At frequencies above 50 kc, it 
becomes practical to use a high-Q tuned circuit for frequency selection. 
Because such a network is resistive only at its resonant frequency and is 
reactive everywhere else, the oscillation condition of zero net phase shift 
can be satisfied only at this particular frequency. Figure 14-14 illus- 
trates how a portion of the developed voltage may be fed back to the 
input of the base amplifier by tapping the tank circuit. Other methods 
of feedback are also in use, and some of these will be discussed below. 

Even if the excitation current is rich in harmonics, the low impedance 
of the tank, everywhere but at its resonant frequency, will not permit the 
development of other than an almost pure sinusoidal voltage. The feed- 
back will further improve the selectivity, and consequently we shall not 
be particularly concerned with the harmonic response. We are, how- 
ever, quite interested in generating a signal whose frequency will not be 
affected by external factors. 

Almost all tuned-circuit oscillators may be represented by the basic 
circuits of Fig. 14-15o and b, where, in order to achieve the proper phase 
relationship with a single-stage amplifier, the tap of the tuned circuit is 
grounded. For simplicity, the biasing and power supplies have been 



446 



OSCILLATIONS 



[Chap. 14 



omitted. In any specific circuit these connections would depend on 
the nature of the three impedance elements. 

We might note that the network of Fig. 14-15 includes two modes of 
feedback, a direct transmission path from the plate (collector) to the grid 
(base) through Z 3 and the mutual coupling linking the input and output 
loops. Some oscillators use purely capacitive elements for Z\ and Z 2 , 
and in others two isolated coils are used. When we treat these cases, M 



f 


± 












r 


n 








\ 






\ 


L / 






"«. 












!>< 










z> 




z 2 






Zx 




Z 2 
















^■M-" 










^-M-^ 








z 3 






z 3 
















(c 


<) 




(b) 






>r p 


~* 




i— S— vvv 


r d &A_ e 


8 


fCXM 






' 






\ 
e 


h > 






r >iN 


e 


( 






Z x 


V 


Z z 






Zi 




z 2 












<^3__ 


^AT-^ 


-> 






C's 


^~~M-^ 


> 














Z 












3 





















(c) (d) 

Fig. 14-15. Basic tuned-oscillator circuits, (a) Using a vacuum tube; (6) using a 
transistor; (c) vacuum-tube model; (d) transistor model. 

would not appear. However, many of the more commonly used circuits 
do contain coupled coils, and therefore M must be included in any general 
treatment. Figure 14-15 also shows the various models with which we 
shall be concerned in the analysis of this type of oscillator. In the 
interests of simplicity, the small reverse-voltage-transmission term has 
been omitted from the transistor model. If this term ever becomes 
significant, it can always be included by inserting a voltage generator 
h re v ce in series with r' n . 

Since the single tube or transistor introduces an effective phase shift 
of 180°, there must be an additional 180° contributed by the network, 
composed of Z\, Zi, and Z 3 . These models may, of course, be treated 
as before by evaluating A/3, where A is the gain when the output is loaded 
by the input impedance of the feedback network and is the transmission 
from the plate (collector) to the grid (base). 



Sec. 14-4] almost sinusoidal oscillations 447 

By observing that the criteria for self-sustained oscillation are inherent 
in the vanishing of 1 - 0(u) A or 1 - T(o>) A c in Eqs. (14-2) and (14-18), 
we can consider an alternative method of obtaining these criteria. The 
nature of the over-all response of any system may be found by solving 
a set of mesh or node equations. This response is undefined when the 
system determinant vanishes. Thus, by writing the loop equations, or 
where appropriate the node equations, finding the system determinant, 
and setting it equal to zero, we satisfy 

1 - 0(a) A =0 or 1 - T(w)A c = 

The vanishing of the imaginary part of the determinant gives the fre- 
quency of oscillation. From the real part, we obtain the circuit condi- 
tions which must be satisfied if the oscillator is to be self-starting. 

Vacuum-tube Tuned Oscillators. For the vacuum-tube circuit of 
Fig. 14-15c, the two loop equations are 

(r p + Zi)U — (Z 2 + Z m )i 3 = —iiei 
- (Z 2 + Z m )i* + (Zi + Z 2 + Zz + 2Z m )i* = (14-19) 

One additional denning equation is needed before the set of equations in 
(14-19) can be solved. The control voltage must be expressed in terms 
of the loop currents. 

«x = -Z m i 2 + (Z m + Zi)i, (14-20) 

Substituting Eq. (14-20) into Eq. (14-19) and collecting terms yields 
the system determinant, 

r p + Z 2 — y.Z m — (Z 2 + Z m — nZ m — p.Z%) 
-(Z, + Z m ) Z 1 + Z i + Z 3 + 2Z m 



A = 



(14-21) 



If we initially assume that the impedance elements are purely reactive 
(Zi = jX t ), then the imaginary portion of the determinant will contain 
only odd powers of Xi. The sign of the reactance is assigned to the ele- 
ment (plus for inductance and minus for capacity). It follows that the 
frequency of oscillation will be given by 

Im A = r„(Xi + X 2 + X 3 + 2X m ) = (14-22) 

Equation (14-22) can be satisfied only when the sign associated with 
one reactance is opposite to that associated with the other two. We 
shall see below [Eq. (14-23)] that Zi and Z 2 must be that same type of 
element to satisfy the gain condition. Thus if Zi and Z 2 are inductive 
at the oscillation frequency, Z 3 must be capacitive, and vice versa. 

Since the sum of the loop reactances vanishes in Eq. (14-22), the real 
part of the determinant of Eq. (14-21) may be expressed as 

Re A = M (X, + X m )(X 2 + X m ) - (X, + X.)» 



448 oscillations [Chap. 14 

By setting this equal to zero, we find that the threshold condition for 
self-starting is 

M > y*tf" (14-23) 

Ai -f- A m 

Equation (14-23) can be satisfied only when Zx and Zi are the same type of 
component: either both are capacitive or both are inductive. If they 
are not the same, the feedback would be negative rather than positive. 

Unless the ratio of impedances is somewhat less than /u, the over-all 
loop gain will not be greater than unity and the oscillator will not be self- 
starting. Furthermore, with threshold operation, the system is very 
sensitive to changes in the parameters of the active element. On the 
other hand, if this ratio becomes much less than m, the limiting may 
introduce an excessive amount of harmonics, with a corresponding 
deterioration of the waveform. To avoid both problems and to have 
some latitude for component tolerances, at least one of the elements 
determining the threshold point should be made adjustable. 

The following possibilities arise where the major mode of feedback 
is through X 3 and the mutual coupling either does not exist at all or is 
purely incidental. 

1. Colpitts oscillator (Fig. 14-16o) 
Xi and Xi are capacitors. 

Xt is an inductance. 
X m = 0. 

2. Hartley oscillator (Fig. 14-16&) 

Xi + Xi + 2X m is a single tapped coil. 
' X* is a capacitor. 

3. Tuned-plate tuned-grid oscillator (Fig. 14-16c) 

Xi and Xi are tuned circuits adjusted to be somewhat inductively 

tuned slightly below a> . 
Xt is the stray grid-to-plate capacity. 
X m = 0. 

When the major mode of coupling is through the mutual inductance and 
Xz represents the insignificant parasitic coupling, the following two cir- 
cuits are also feasible: 

4. Tuned-plate oscillator (Fig. 14-16d!) 
Xi is a tuned circuit in the plate loop. 
Xi is the grid coil. 

2X m is the mutual coupling between the two coils. 

5. Tuned-grid oscillator 

Virtually identical with the tuned-plate oscillator except that the 
tuned circuit is moved into the grid loop. 



Sec. 14-4] 



ALMOST SINUSOIDAL OSCILLATIONS 



449 



Various hybrid oscillators combining significant features of the five 
listed above are also in use. The above tabulation only indicates the 
range of possibilities; it makes no attempt to exhaust them. 





(°) 



(b) 




'At 



M 



— L\\\ 1 ( ) C, i L 5 o 



T 



(c) (d) 

Fig. 14-16. Some practical vacuum-tube tuned oscillators, (a) Colpitts circuit; (6) 
Hartley circuit; (c) tuned-plate tuned-grid; (d) tuned-plate oscillator. The various 
components not labeled are used for bias and decoupling and would be chosen to have 
a negligible influence on the frequency of oscillation. 

In the Colpitts oscillator of Fig. 14-16a, the frequency of oscillation as 
found from Eq. (14-22) is 



woe 



where C T = C1C2/C1 + C 2 . For self-starting, n > Ci/Cj [from Eq. 
(14-23)]. The Hartley circuit (Fig. 14-166) oscillates at 



-J 



1 



and here 



M> 



Cz^ + U + 2ikf) 

Li + M 
Li + M 



In both circuits the frequency of oscillation is simply the resonant fre- 
quency of the LC combination. The capacitive term in the Colpitts 



450 oscillations [Chap. 14 

equation is the series combination of C x and C 2 . If the tube has a 
reasonably large gain, C 2 <C Ci and Ct = C 2 . In the Hartley circuit, 
the inductive term is the total tank inductance and, with a high-gain 
tube, L2 » L\. 

The Hartley and the two single tuned-circuit oscillators (circuits 2, 
4, and 5 above) are best suited for variable-frequency operation. In 
these, the frequency may be changed by varying a single capacitor. 
Moreover, this adjustment will not affect the conditions for self-starting 
operation, which depends on the mutual coupling between two coils and 
on the location of the coil tap. In the other oscillators either two LC 
circuits must be retuned or two capacitors must be simultaneously 
tracked. 

If the tank elements are dissipative, or if they are loaded by the external 
circuit when coupling to the next stage, then the associated resistance 
will change both the frequency of oscillation and the conditions for 
self-starting. Provisions for retuning and for adjusting the amount of 
feedback allow us to compensate for these changes. There is no point in 
calculating the new condition for threshold operation. It is, however, of 
some interest to recalculate the frequency of oscillation. By doing this 
we shall see that the circuit dissipation also makes the frequency depend- 
ent on the parameters of the active element. 

Example 14-1. Let us consider, for example, the basic Colpitts oscillator of Fig, 
14-16a, where the nominal unloaded frequency is w = 10 7 radians/sec. The tube 
used has y. = 20 and r, = 5 K. The tank is so designed that Q \/L/Ct — 25 K and, 
in addition, its loaded Q — 25. 

From the above conditions, the tank components are found to be Ct = 100 ppf and 
L = 100 iih. To ensure self-starting and to allow for changes in the tube parameters 
we shall satisfy 

" = 2 C 2 

instead of the threshold condition. Solving yields Ct = 110 npi and Ci = 1,100 ii/d. 
The final parameter — the equivalent series coil resistance — is 

10 7 L 10' X 10-" .„ , 
r = .. = ^ = 40 ohms 

By substituting Zz = r + jaL, Zi = —j/aiCi, and Z% = —j/aCz into Eq. (14-211 
and collecting the imaginary terms, the frequency of oscillation is found from 



The frequency becomes 



■-7T-0 



"' \ LC,C 2 \ ' r Cy + Ct r v ) 



\ ^ 1,210 5,000/ 



Sec. 14-4] almost sinusoidal oscillations 451 

or the frequency increases by approximately 0.36 per cent over its nominal value as a 
direct result of the resistive loading of the tank. The conditions for self-starting will 
also change; this calculation is left as an exercise for the reader. 

Unless r p S> r, the circuit would be quite sensitive to the changes in r p 
occurring during the life of the tube or when replacing tubes. Any resis- 
tive loading of the tuned circuit would effectively increase r and would also 
change the frequency of oscillation. External loading may be minimized 
by using an amplifier to decouple the final load from the oscillator stage 
and by loosely coupling this stage to the tank. 

Transistor Tuned Oscillators. From the denning equations for the 
transistor model of Fig. 14-15d, the over-all system determinant is found 
to be 

r' n + Z X Z m - (Z, + Z m ) 

A= pu + Z m n + Z t -(Z* + Z m ) (14-24) 

-(Zi + Z.) ~(Z» + Z m ) Zx + Z 2 + Z 3 + 2Z„ 

When the impedances are all purely reactive, the imaginary part of the 
determinant will reduce to 

v{,(Ii + X 2 + X 3 + 2X m ) - X 3 (X 1 X 2 - I« ! ) - (14-25) 

where the reactance term includes an associated sign. We might note 
that the first part of Eq. (14-25) contains the total series impedance of 
the self-resonant circuit composed of the three reactive elements. The 
last portion is a correction term due to the nature of the external loading 
of the tank. In general, we shall have to minimize the influence of these 
external factors if the oscillator is to have good frequency stability. 

The significant portion of the frequency-determining equation is 
identical with that found for the vacuum-tube circuit. As discussed 
above, the nature of Z 3 must be opposite to Z\ and Z 2 . Consequently 
the possible circuit variations are those previously tabulated. 

By assuming that the actual frequency of oscillation is very close to 
the natural frequency of the tank, the term 

X-, + X 2 + Xz + 2X m £* 

and the determinant may be simplified before solving for the condition 
for self-starting. It shall be found by setting the real part of the remain- 
ing terms equal to zero. 

r' u (X 2 + X m y + r„(Xi + X m y - 0n(Xx + X m )(X 2 + X m ) = (14-26) 
Because the second term on the left-hand side is relatively insignificant, 
Eq. (14-26) may be further simplified. The resulting criterion for self- 
starting oscillations becomes 

A 2 -f- X m - pTd re_ f14— 97^ 

v i Y — 7' r' V 1 ^ - *'^ 

■A-l -\- A m r u r n 



452 



OSCILLATIONS 



[Chap. 14 



To justify the use of this equation, we shall solve Eq. (14-27) for (Xi + 
X m ) and substitute back into the original expression. 



rii(x, + x m y + 



(r'nY 



(X. + X„) 2 - rJ 1 (X 2 + X.)« = 



The neglected second term is only rJ 1 /j8V < j times as large as the first term. 
Since r[ x is the small input impedance, ^Vj ^> r' n and the condition 
expressed in Eq. (14-27) is essentially correct. 





(a) (b) 

Fig. 14-17. (a) Transistor Colpitts oscillator; (b) Hartley oscillator. 

From Eqs. (14-25) and (14-27) the frequency and gain requirements 
of the Colpitts oscillator of Fig. 14-17o are 



woe 



\ LCxC 



C-l . 1 



where \Z(Ci + C 2 )/LCiCt is the resonant frequency of the unloaded 
tank. For self-starting, the Colpitts oscillator must be adjusted so that 

£c > Ci 

r'n ~ C 2 
The Hartley circuit of Fig. 14-166 oscillates at 

1 



UOH — 



4 



and for self-starting, 



C(Li + U + 2M) 



r. ^ Lt + M 
r' n -Lt + M 



L\Li 



M* 



rjru 



In both cases the resistive loading of the tuned circuit causes a small 
change in frequency from the nominal resonant point. This can be 



Sec. 14-4] almost sinusoidal oscillations 453 

minimized by maximizing the rjr'u product and by properly choosing 
the ratio of L to C. Furthermore, by using tightly coupled coils in the 
Hartley circuit, the numerator of the correction term will be reduced; 
with unity coupling it will reach zero and the circuit will oscillate exactly 
at o>o. 

Crystal, Oscillators. A suitably clamped quartz crystal may be made to 
resonate by exciting it with an electrical 

signal of the proper frequency. The T 

mechanical oscillations are fixed by the ■ 
relative dimensions of the crystal, and the | 
damping depends on the characteristics of | — | 
the mounting. As the damping factor can ~~ T~ 
be made quite small, the electromechanical I 
Q will be proportionately large; expected I 

values are between 1,000 and 50,000. To ^ 14 _ lg ^^^ electrical 
use the crystal in a fixed-frequency oscil- circuit of a quartz crystal, 
lator, we simply substitute it for one of 
the reactances in the basic tuned-oscillator configuration of Fig. 14-15. 

Sinusoidal oscillation is possible at one of the two modes of electro- 
mechanical resonance corresponding to the electrical resonances of the 
equivalent circuit of Fig. 14-18. The parallel-resonant mode depends on 
the mass factor L t and the capacity of the mounting plates using the 
crystal as a dielectric, C. At this frequency, the terminal impedance is 
purely resistive and is extremely large. In its other mode, the mass and 
spring constant of the crystal, L,- and C,-, are series-resonant. Here the 
crystal is essentially a short circuit. Since the dissipation term r,- is 
small, the network appears reactive everywhere but at these two fre- 
quencies. Many crystals are cut to have several resonant points, and 
these might be incorporated into the electrical model by adding, in 
parallel, additional LfiCi series arms. 

Even though the frequency of oscillation is essentially that of the 
crystal, minor adjustments are possible, without reducing the Q. By 
adding series capacity or inductance the frequency can be changed by a 
few parts in a million. For wider ranges, the variable-frequency circuits 
would be used instead. 

Two crystal-controlled oscillators are shown in Fig. 14-19. In the 
Pierce circuit, the crystal controls the amount of feedback applied. At 
its series-resonant frequency, the low impedance presented increases 
the amount of feedback to a point where oscillations can be sustained. 
Since the grid circuit is capacitive, the crystal must be slightly inductive 
for the proper phase of the feedback signal. To satisfy the conditions 
found above, the plate tank would also be tuned somewhat capacitively. 
To all other frequencies the crystal presents an extremely high reactance. 



454 oscillations [Chap. 14 

It thus reduces the loop gain well below the threshold value. Further- 
more the net phase shift of the feedback voltage will no longer be the 
180° needed for oscillation. 

The second circuit of Fig. 14-19, the transistor oscillator, is a tuned- 
input tuned-output circuit with capacitive feedback. Here the output 





(a) (b) 

Fig. 14-19. (a) A Pierce oscillator using the crystal as a series-resonant mode; (6) 
a transistor oscillator where the crystal operates as a parallel-resonant circuit. 





MA 




4HH 






1-megacycle 

crystal 

oscillator 


Buffer 

amplifier 

and clipper 


Differentiator 








i 


_ 


^ 


_ 


1-kc 
multi 


10-kc 
multi 


100-kc 
multi 


) 

100-,«sec 
period 


10-^sec 
period 


J, 








1 1 1 



lv<sec period 



Fig. 14-20. A crystal-oscillator-controlled system designed for precise timing measure- 
ments. 

will be developed when the crystal is a high impedance, i.e., at its parallel- 
resonant frequency. At all other frequencies the small reactance pre- 
sented will load the amplifier and will also cause the feedback voltage 
to be in quadrature to that needed for oscillation. 

Crystal-controlled oscillators are widely used as frequency standards. 
By amplifying, clipping, and finally differentiating a 1-megacycle sinus- 
oid, we derive a train of negative pulses spaced 1 Msec apart. As shown 
in Fig. 14-20, these are next applied as a trigger source to synchronize 
an astable multivibrator normally operating at 100 kc. The 10-^sec- 
period pulse train may be further divided down by the other multi- 



Sec. 14-5] almost sinusoidal oscillations 455 

vibrators shown. From this chain we derive pulse outputs spaced by 
10 jusec, 100 jisec, 1 msec, etc. Since the base frequency can be main- 
tained to within 1 part in 10», this system has been used for highly 
precise timing. 

14-5. Frequency Stabilization. All oscillators are to some degree sus- 
ceptible to frequency changes caused by the resistive loading of the 
selective network. If the loading were to remain constant, then the 
initial calibration of the instrument would correct for the deviation from 
the nominal tank resonance frequency. Unfortunately some of the dis- 
sipative elements may be expected to change with time, sometimes in one 
direction and sometimes erratically. All the circuit components con- 
tribute to this drift; even slight changes in the power-supply voltage 
will shift the operating point and thus affect the parameters of the 
active element. Therefore any oscillator which is designed as a precise 
frequency standard should be compensated with respect to all such 
variations. 

We implicitly assume that the nature of the elements comprising the 
selective network is such that they themsebes are completely stable with 
respect to aging and ambient-temperature variations. 

Since it would be identical with detuning the tank, we are also neglect- 
ing the effects of any reactive loading of the tuned circuit. Some react- 
ance would always appear across a portion, or across the complete tuned 
circuit, because of the interelectrode and stray capacity associated with 
the tube and the transistor. Choosing a large tuning capacity would, 
of course, make the frequency much less dependent on any changes in the 
parasitic-circuit elements. 

Our first objective, then, is to eliminate, as far as possible, any other 
factors producing a frequency shift away from <o . Even though special 
provisions may be made in each individual circuit, such as using tightly 
coupled coils in the Hartley oscillator, in Sec. 14-4 we saw that the per- 
formance of all oscillators will be materially improved by minimizing the 
external loading of the network. First, a buffer amplifier may be used to 
decouple the oscillator from the varying output load, and second, since 
the harmonics produced in the limiter are reflected as additional loading, 
the circuit can be adjusted close to the threshold of oscillation. Finally, 
we should investigate the mechanism of frequency instability with a view 
toward selecting the circuit which best suits our particular needs. 

Stabilisation Factor. The frequency of oscillation is identically that 
frequency at which the net phase shift around the amplifier and feedback 
loop is zero. Since the feedback network contains reactive elements, the 
insertion of any resistance will change the phafie response and also the 
null frequency. For example, assume that a change in one parameter, 
external to the tuned circuit, produces a net phase shift of 5° at the 



456 oscillations [Chap. 14 

nominal frequency of oscillation /<>. For sustained oscillations, the 
remainder of the circuit must contribute the complementary shift of 
— 5°. If the normal rate of phase variation in the vicinity of / is — 10° 
per 1,000 cycles, then a frequency increase of only 500 cycles will reestab- 
lish the condition of zero net phase shift. The new frequency of oscilla- 
tion would be /o + 500. From this simple example, we conclude that 
the highest degree of stability would correspond to the most rapid change 
of phase with frequency. 

If, in an oscillator, we can find one set of elements whose phase varies 
most rapidly with frequency, then these elements would have the most 
pronounced effect on the over-all frequency stability. This will, in gen- 
eral, be the frequency-selective network. As a measure of frequency 
stability, we define the normalized sensitivity factor 

S ' S A^To (14 - 28 > 

The larger the value of S f , the more stable the oscillator with respect to all 
changes in the circuit which may produce an undesirable frequency shift. 
When, in the limit, S f becomes infinite, the frequency of oscillation is 
completely independent of all other sections of the system. 

Figure 14-21o is a plot of the phase behavior of a Wien bridge. In the 
vicinity of its null frequency, the response is similar to that seen in all 
other unbalanced null networks (Fig. 14-216). For this particular exam- 
ple, the phase of $', as found from Eq. (14-14), is 

(ii_-o\_ tan _ 1 l/«_« \ 
\wo « / 3 \coo a / 

This is plotted as a function of &>/&>o for three different degrees of unbal- 
ance. One of the three is the completely unbalanced bridge (8 = 3). 
It is readily apparent that Sf reaches a maximum at &> and that the 
closer the bridge is to balance, the larger its value. Differentiating Eq. 
( 14-29) with respect to to and multiplying by co yields 



^ = tan J — <j — 



Sf = Wo 



3 - 8 

d4 9__ 

da 



1 + 



( 3 — & \ 2 /<w _ woV ill (j± _ fjJoV 
9 / \a>o «/ 9 \<«io w/ 



Since 8 > 3, both terms will have the same sign and each, will reach its 
maximum at a = wo- Evaluating S/ at this frequency, 

S f . = -%« (14-30) 

The minus sign indicates the direction of the phase change with frequency. 



Sec. 14-5] 



ALMOST SINUSOIDAL OSCILLATIONS 



457 



Equation (14-30) shows that the bridge improves the circuit stability in 
proportion to its degree of balance. We might note that previously we 
had shown that this same condition was necessary for the optimum 
harmonic rejection [Eq. (14-16)]. 









1 1 

8-300 





































































6 


-3 


3 s 






















































































































6=3 






































































X. \, $ 


-30 


































5-300"*"^. 























































150" 

120 

90 

60 

30 ^. 
8 
I 
-30 

-60 

-90 

-120 



-150° 



8 10 



(a) 



a 




+90° 



-90' 




Twin T and 
Bridged T 



(6) 



\tf\ 



V3 for the Wien Bridge 
1 for the Twin T and 
Bridged T 




(c) 



Fio. 14-21. Amplitude and phase characteristics of null networks, (a) Phase response 
of the Wien bridge; (b) the phase response of unbalanced bridge using a twin-T or a 
bridged-T network for one pair of arms; (c) the general amplitude response of the null 
networks. 

Unless a sharp null is exhibited by the network, the phase changes 
relatively slowly in the vicinity of « . The curve for S = 3 in Fig. 14-21a 
illustrates a variation of this nature. Since this curve represents the 
behavior of a completely unbalanced bridge, it also typifies the response 
of such EC and RL circuits as the phase-shift oscillator. 

The phase response of a tuned-circuit oscillator is related to the 
circulating current through the three reactive components Z\, Z t , and 



458 



OSCILLATIONS 



[Chap. 14 



Z%. Near a> the relative phase of the feedback voltage, which is devel- 
oped at the active-elements input, may be approximated as 



4> = tan -1 Qo 






(14-31) 



If this were to be plotted, we would have a family of curves similar to 
those drawn for the Wien bridge. After all, a tuned circuit is also a 




Fig. 14-22. Meacham-bridge oscillator. 

null network. The phase starts at 90°, reaches the 180° necessary for 
the positive feedback at to , and then continues increasing toward 270° at 
the higher frequencies. The frequency-sensitivity factor of this phase 
response, evaluated at o>o, is 

5,. = 2Q 

For optimum stability we should use a high-Q circuit and should set 
the oscillation frequency as close to co as possible. An extremely good 
choice for one of the elements is a crystal which will give a frequency- 
stability factor of 10 s to 10 6 . With a normal tuned circuit, the best 
that can be expected is about 10 2 . 

For an extremely precise frequency standard we can combine the 
improvement found in a high-Q crystal-controlled oscillator with the 
selectivity multiplication of an almost balanced bridge. The Meacham 
oscillator of Fig. 14-22 has the highest stability of any circuit yet devised. 
The bridge nulls at exactly the series-resonant frequency of the crystal. 
At this frequency the crystal appears completely resistive, which makes 
the bridge easy to balance. By using a small degree of unbalance (1/5) 
and approximately equal resistance arms, the stability factor will be 

S f . - 2SQ 

and for Qo = 10 6 , stabilization up to 1 part in 10 8 is possible. This 
means that if the amplifier introduces a phase shift of 0.1 radian (a 
relatively large value), the frequency would change by only 1 part in 10 9 



Sec. 14-5] almost sinusoidal oscillations 459 

Reactive Stabilization. Llewellyn showed that it is possible to stabilize 
an oscillator by inserting reactances in series with the terminals of the 
active element or in series with the applied load. The theory behind 
such stabilization is that the tank circuit appears resistive at its own 
resonant frequency and that only the need to compensate for the addi- 
tional loop phase angles causes the frequency to shift. Now if the 
external resistance sees a completely resistive internal impedance, it 
would not affect the phase of the system; it would only increase the load- 
ing and the gain required for threshold operation. At many points in the 
circuit the two-terminal impedance has a reactive part. When this is 
loaded, the complete phase response of the system changes. By inserting 
in series with the external load a reactance of the opposite type to that 
originally seen, it is possible to tune the subsidiary loop and thus reduce 
the net phase shift to zero. The load now sees a pure resistance, and the 
frequency becomes independent of the load and of load variations. 

The nature of this stabilizing reactance depends on the terminal at 
which it is introduced and on the configuration of the over-all network. 
It is also possible, by inserting a single reactance at an appropriate point, 
to compensate for the loading at several different parts of the network. 
But in general, compensation elements would have to be inserted in 
series with each of the resistive components of the external circuit. 

As an example, we shall insert such a reactance in series with the 
base in the Hartley transistor oscillator described in Sec. 14-4. For 
simplicity, the mutual coupling will be taken as zero. Furthermore, since 
the stabilization will return the frequency to cj , the series impedance 
of the tank will also be zero. The simplified system determinant will 
now be 

r'„ + Z. + Z 1 -Z 1 



A = 



Pn Z% + Ti —Zi 

-Zx -z* 



where Z, represents the stabilizing reactance. Setting the imaginary part 
equal to zero (when all impedances are purely reactive) yields 

X.Xf + X 1 X 2 i + X 1 i X i = 

and this equation enables us to find the value of X,. 



'•--^O + fi) 



But if the active element employed in the oscillator has a high gain, 
then, from Eq. (14-27), we see that 1 » Xi/Xt and the stabilizing react- 
ance 

X. S - Zi (14-32) 



460 oscillations [Chap. 14 

The element to be inserted in series with rj x is a capacitor whose reactance 
at o) is given in Eq. (14-32). 

In order to obtain an order of magnitude for C„ consider a circuit 
where C = 250 md, L x = 10 nh, L 2 = 990 jah, and « = 2 X 10 6 . Sub- 
stituting into the expression for X, yields 

C ' = 5?Li = 4 X 10+ 12 X 10 X 10-« = 0025 " f 

This value is quite feasible, and it would appear in the circuit as an 
input coupling capacitor. Note that C, depends on the frequency and on 
the value of L\. For variable-frequency oscillators, stabilization of 
this nature is obviously impractical. 

14-6. Amplitude of Oscillations. The exponential build-up of the 
self-starting oscillator will continue as long as the average gain over 
the cycle is greater than the threshold value. Eventually, the amplitude 
of the output sinusoid is limited as its peaks force the instantaneous 
operating point into the nonlinear regions. As a crude first-order 
approximation, we can say that the output will be equal to the linear 
capabilities of the system. Since this depends on the nature and point of 
application of the nonlinearities, we shall now examine several of the 
amplitude-limiting schemes used in practical oscillations. Fortunately, 
the exact signal level is much less significant than the exact frequency, 
and therefore we can justify relatively gross approximations in treating 
this problem. 

1. Active-element Limiting. The first case considered is where the 
amplitude is limited by the tube (transistor) being driven into saturation 
and/or cutoff. We shall make the following assumptions as to the 
oscillator behavior. 

1. The gain is adjusted to slightly beyond the threshold value so that 
the output is not excessively distorted. 

2. The selectivity of the frequency-determining network will greatly 
reduce the amplitude of the harmonics produced, and consequently there 
will be an almost pure sinusoid present at one point at least. 

Since the response is sinusoidal at one point in the closed system, 
the oscillator behavior may be simulated by opening the loop at this point 
and exciting the open circuit from an external sinusoidal source of the 
same frequency. For example, in the Colpitts oscillator of Fig. 14-23o, 
the loop has been opened at the grid. To avoid changing the over-all 
response, the network must be loaded by the impedance it normally sees. 
•The sinusoidal driving signal must also be injected through a source 
impedance of the appropriate magnitude. 

In our effort to describe the nonlinear behavior of the oscillator, with- 
out having to account for the different impedances presented by the tank 



Sec. 14-6] almost sinusoidal oscillations 461 

at its various harmonic frequencies, we shall assume that the plate load 
is a constant equal to the resistance of the tank at resonance. This 
incorrect assumption can be justified only by noting that, in an almost 
sinusoidal oscillator, the fundamental component is the predominant 
term. Any error resulting will be insignificant if the totality of the 




d=C 2 Cn d= e f -0 o e b 



(a) 




AE., 



E P t 



Slope.- t; 



-E P i 
~AE s i m 

(b) (c) 

Fig. 14-23. (a) Open-circuit Colpitts oscillator; (6) plate-circuit operating path at the 
fundamental frequency; (c) plate waveshapes for two values of drive, assuming a 
constant-resistance plate load. 





/ \ 




■M ) 






9\2 It 2¥ 


2n N 








k <Oot 


\ / 
- - ^_^ 




V. 



harmonics calculated on the basis of a constant-resistance load remains 
small. But this is the same as saying that the net gain in the small- 
signal region should be adjusted to slightly beyond the threshold value. 
By solving Fig. 14-23 we find that the effective input resistance of the 
unloaded high-Q tuned circuit at resonance may be expressed as 

B - = B = (crTc s ) 2< ^ (14 - 33) 

From the self-starting condition for this oscillator, C% < pCi, we see 
that Ct is the smaller capacity; when /*• > 10, it will completely pre- 



462 oscillations [Chap. 14 

dominate. Thus the nominal resistance of the tank at resonance reduces 
to 



R = Q \c; 

Using the equivalent load resistance R, the path of operation of the 
base amplifier is constructed (Fig. 14-236) and the bounds of the linear 
region delineated. 

For an input excitation of 

e, = E,im cos a>ot 

the output remains sinusoidal up to the linear capabilities of the tube 
(point x or y in Fig. 14-236 and the corresponding point W in Fig. 14-24). 
Above this value the output is clipped by the nonlinearity as shown in 
Fig. 14-23c. In reality, the tank cannot sustain such a waveshape, but 
from it we can determine the approximate amplitude of the fundamental 
component of the tank voltage. Equation (14-34) may be used: 

1 /" 2ir 
Epi™ = ~ I e p cos uot dwot (14-34) 

t Jo 

where e p is the time- varying component of the plate voltage. When the 
output is not readily expressed in the form of an analytic function, the 
fundamental will be found by a graphical integration or by a schedule 
method. 

Suppose, for instance, that the tube is biased so that the plate operating 
path is symmetrical about Ebb- One clipping point is at the E c = line, 
while the other corresponds to cutoff. If we further assume that the 
limiting in both regions is ideal, then the approximate waveshapes with 
which we are concerned are shown in Fig. 14-23c. From Fig. 14-236, 
the peak signal in the linear region of operation is 

E p i = Ebb — Ebi = A(E B i m )i = ■= — , p Ebb (14-35) 

where A = —nR/(r p + R). In a transistor oscillator biased in the 
center of its active region E p i = Ebb- 

Because of the symmetry exhibited by the signal of Fig. 14-23c, 
Eq. (14-34) reduces to 

4 /"» 4 f r/2 

Epi m = - / E P i cos <i>ot dwot + - I AE. lm cos 2 wot daat 
ir Jo r Ja 

= - E Pl [sin 6 + ^^ (-2 2- ) J (14 " 36) 

where 6 = cos -1 — ~ 



AE, 



Sec. 14-6] 



ALMOST SINUSOIDAL OSCILLATIONS 



463 



The function describing the amplitude response of the oscillator, that 
is, E p i„ versus E, Xm , may now be plotted (curve a in Fig. 14-24). This 
same describing function is often obtained experimentally, by measuring 
the output as the input excitation is increased. At each point on the 
curve the ratio of E pim /E, lm is the average gain of the base amplifier over 
1 cycle of the fundamental. Since the criterion for the stable oscillation 
is satisfied when the gain is equal to 1/jS , by superimposing a line having 




Fia. 14-24. Curve a, describing function for the oscillator of Fig. 14-23; curve 6, 
describing function of a non-self-starting oscillator. 

this slope and by locating its intersection with the describing function, we 
find the amplitude satisfying 



A-v — 



E. 



E,\ 



J. 

00 



When the small-signal gain is only slightly greater than the threshold 
value, point Z will fall close to the upper limit of the linear range of the 
circuit. Thus the area between the l//3 line and the describing func- 
tion gives a qualitative measure of the harmonic content of the output; 
if the area is small, the signal will be an acceptable sinusoid; if it is large, 
the output will be somewhat distorted. That Z of Fig. 14-24 is the sole 
stable point may be verified by considering the response to a small 
change in the input excitation. If the input is momentarily reduced, 
then the new value of loop gain becomes greater than unity, indicating 
that the signal must now increase with time. Above point Z, the average 
loop gain is less than the minimum value needed to sustain the oscilla- 
tions and they begin decreasing toward the stable value. 

Curve b in Fig. 14-24 illustrates the shape of a describing function of 
a non-self-starting oscillator. This might be the circuit of Fig. 14-23 
when the tube is biased on the verge of cutoff instead of in the center of 
its linear region. At small-signal levels the average gain is less than 1//S 



464 



OSCILLATIONS 



[Chap. 14 



and the circuit is unable to sustain oscillations. However, if some 
external excitation forces the operating point past S, the oscillatory 
criterion is satisfied and the amplitude will continue to increase toward 
point Z'. 



, .„ AE, lm cos8,.~ 




(a) 





Fig. 14-25. (o) Grid-bias-limited Colpitts oscillator; (6) plate waveshapes for small 
a and large 6 excitation; (c) describing function for a circuit with bias limiting. 

Nonsymmetrical clipping of the sinusoid will produce an additional 
d-c component in the plate current. It would be found from 



2t Jo 



R 



dut 



and since 7 d0 flows through R K , the bias would change slightly. The 
resultant shift in the transfer characteristics might have to be considered 
when constructing the describing function. This shift is in the direction 
of more symmetrical operation, and it therefore aids in reducing the 
harmonics present in the output. 

2. Bias Limiting. The general circuit response is similar to that 
considered in case 1, except that the bias is derived by the grid current 
charging of C„ (Fig. 14-25a). The only difference in the analysis is that 



Sec. 14-6] almost sinusoidal oscillations 465 

the input sinusoid would be clamped at zero by the energy-storage ele- 
ment included in the input lead. Consequently the position of the load 
line shifts with the driving-signal amplitude. The locus of the minimum 
plate voltage is the E„ = line. As illustrated in Fig. 14-256, until such 
time as the grid is driven below cutoff, the output voltage will vary sym- 
metrically about E». 



-^*^4fc^-4p^ 



Fig. 14-26. Intermittent oscillation in a grid-bias-limited oscillator. 

E P i m is readily evaluated once 6 is defined. From the clamped grid- 
circuit waveshape of Fig. 14-256, 

e c = —E.i m + E. lm cos &xrf 

Hence is given by 

, = cos _ 1 ( 1+ | f J 

_ E» + AE, lm cos 9 

where &«, — ~ 

/* 

For this type of limiting, the amplitude would also be found from the 
describing function of the system (Fig. 14-25c). Of particular interest 
is that the output can reach a maximum and then decrease with an 
increasing input. When the output amplitude is stabilized on the 
. decreasing segment of the describing function, the circuit is prone to 
intermittent oscillations, or "squegging." On the positive peaks the 
large pulses of grid current charge the input capacitor. If the net charge 
accumulated in C„ on the positive peaks is greater than the energy 
dissipated in B s over the remaining portion of the sinusoid, the bias 
voltage will build up to a point where it cuts off the tube. The larger 
the net charge accumulating, the fewer the number of cycles contained 
in the oscillation interval. As the oscillations damp out (Fig. 14-26), 
C recovers toward zero, with the long time constant C g R g . The tube 
eventually turns on, and the cycle repeats. Reducing the time constant 
will often allow the circuit to oscillate properly. 

3. Feedback Limiting. In this case, a passive nonlinear element is 
included in the feedback network at a point where it can change the 
magnitude of (3 but where it will not affect the null frequency. For 



466 



OSCILLATIONS 



[Chap. 14 




Fig. 14-27. Amplitude-controlled Wien- 
bridge oscillator. 



example, in the Wien-bridge oscillator of Fig. 14-27, the degree of bridge 
unbalance is controlled by the thermal characteristics of the tungsten 
lamp, which is used as one of the resistive arms. Its resistance increases 
with temperature, and hence with the voltage applied across the branch 
and across the bridge (Fig. 14-28). Normally the bridge is slightly 
unbalanced (0' = 1/8), adjusted so that the amplifier can just sustain 

the given amplitude of oscillation. 
Any increase in the output voltage 
will increase the resistance of the 
lamp and bring the bridge into 
closer balance, thus increasing S. 
The loop gain will drop below unity, 
and the oscillations will start damp- 
ing out. Any decrease in the out- 
put would establish the conditions 
necessary for an amplitude build-up. 
We conclude that this form of re- 
sistive limiting maintains a con- 
stant output amplitude with a high degree of stability. The normal 
operating condition is now given by 

A0'(e,u) = 1 (14-37) 

where A is simply the gain constant of the system. At o> this reduces to 

S(e) 

and the particular voltage e satisfying this equation is the oscillation 
amplitude. Since the limiting process readjusts the /?' of the network, 
this oscillator will always operate under the threshold condition expressed 
in Eq. (14-37). Moreover, as the amplifier is never driven into its 
nonlinear region, the output will be almost completely free of harmonic 
distortion. 

The lamp has thermal inertia, and consequently it will only respond to 
the mean value of the voltage applied over some small time interval. A 
standard small bulb, operating in the range of 600 to 1000°K, far below 
the temperature used for illumination, has a thermal time constant 
between 20 and 75 msec. The mass of the filament and the construc- 
tion of the bulb are the determining factors; the smaller the nominal 
power rating of the lamp, the shorter the time constant. With respect 
to any single cycle, the tungsten lamp may be treated as a fixed resistance. 
At the lower audio frequencies where the period is comparable to the 
thermal time constant (below 20 cps), the change in resistance over the 



Sec. 14-6] 



ALMOST SINUSOIDAL OSCILLATIONS 



467 



individual cycle distorts the output sinusoid and sets a lower frequency 
limit for this method of amplitude limiting. 

1,200 
1,100 
1,000 

R„ *°° 
40 800 

35 700 

30 600 

25 500 

20 400 

15 300 

10 200 



100 






















^ Mazda 
















// 


f 




6w 
120 v 














// 




















/s 


«-400+45. 


; 
















// 


V 










•'' 


6 


















> 


■z^-*, 


-6.. 

150 

pilot 


Jv 
ma 
amp 






// 








^s' 








/ i 






/£■ 


<^ 


■KM 


14£ 






































''/ 







































































2 4 6 8 10 12 14 16 18 20 E a — *- 

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 E b *■ 

E rms , volts 

Fig. 14-28. Resistance characteristics of two typical tungsten lamps and their linear 

approximations. 

Example 14-2. Let us consider the design of the amplitude-limited bridge of 
Fig. 14t-27. The circuit contains an amplifier which has a nominal gain of 300 and a 
dynamic range of 120 volts peak to peak. The 6-watt Mazda lamp of Fig. 14^-28 will 
be used for amplitude limiting. 

To allow some latitude for variations in the amplifier response, we shall limit the 
output sinusoid to 30 volts rms. With the large gain available, the bridge will be 
almost completely balanced. For threshold operation Si = At — 300, leading to 



«. 



Ri + i?2 



300 



= 0.330 



The voltage across the lamp is 

Est - 0.33S. - 9.9 volts 

Since a high degree of quality .control cannot be expected in a device made for 
illumination and not originally designed as a control element, the actual lamp charac- 
teristics will vary over very wide limits. Any reasonable approximation will serve 
for design purposes; for simplicity we shall use a straight line. Within the range of 



468 oscillations [Chap. 14 

interest (2 < E < 18), the Mazda lamp will have a resistance variation given by 

R, Si 400 + i5E 

At 9.9 volts across the lamp, the effective resistance is 845 ohms. Substituting into 
the bridge arm equation, 



fli 



1.0 - 0.33 
0.33 



Rt = 1,706 ohms 



Ri would be made somewhat adjustable to allow for variation in the lamp resistance. 
We shall now assume that the bridge is balanced as calculated above and that the 
amplifier gain decreases by 33 per cent because of the aging of the tubes. To sus- 
tain the oscillations, R 2 must also change until it satisfies 



R, 



fl„ + fti 



200 



= 0.3283 



Thus Ri becomes 



«, = (!- 0.0076)B 2 = 838.6 ohms 



The decrease of 0.76 per cent in the lamp resistance corresponds to a new lamp volt- 
age of 



E K , 



9.755 volts 



The new stable output is 29.71 volts. A gain reduction of 33 per cent is accompanied 
by an amplitude change of only 1.0 per cent. Consequently, this method of limiting 
also ensures amplitude stability. At the new output, the oscillator again operates 
under threshold conditions. 

The technique of limiting by automatically controlling the feedback 
has also been applied to the Meacham bridge (Fig. 14-22), where the lamp 

replaces R$, and to several tuned- 
circuit oscillators, where a resistance 
bridge containing lamps might be 
used for feedback. Figure 14-29 
illustrates an amplitude-controlled 
bridge used to separate the feedback 
control from the frequency-selective 
network. Limiting may be as- 
signed solely to the bridge. Other 
tuned circuits have used small lamps 
in series with the coil to control the 
effective Q of the tank. 




Fig. 14-29. Bridge-controlled feedback 
oscillator. 



Besides tungsten lamps, most semiconductor materials (e.g., carbon, 
silicon, and germanium) and many compounds, such as silicon carbide, 
exhibit temperature-sensitive resistance properties. All these have been 
used for amplitude limiting. But since the semiconductor thermistors 
have a negative temperature coefficient of resistance, in contrast to the 
positive one of the pure metals, they will be used to replace the comple- 
mentary arm of the bridge. In the Wien-bridge circuit of Fig. 14-27, the 



ALMOST SINUSOIDAL OSCILLATIONS 



469 



Sec. 14-6] 

thermistor would substitute for Ri instead of Ri. Thermistors are 
fabricated in a much wider range of resistance, and rate of resistance 
variation with temperature, than can be expected from a tungsten 
filament. Operating as they do at a lower temperature (300 to 375°K), 
they are extremely sensitive to changes in the ambient. For acceptable 
amplitude stability some form of temperature compensation would be 
necessary. 

4. Automatic Gain Control. Threshold operation of the oscillator 
may also be ensured by using the output amplitude to control the gain 
of the base amplifier. With many active elements it is possible to find 
some d-c voltage or current which 
will determine the magnitude of the 
forward transmission. For exam- 
ple, in a pentode the grid-to-plate 
transconductance is a function of 
the suppressor voltage. Remote- 
control tubes are designed with a 
gain that decreases with the control- 
grid bias. In a tetrode junction 
transistor the value of /3 may be 
adjusted within relatively wide 
limits by injecting a bias current 
into the second base connection. 




^*%ny 



Fig. 14-30. Basic automatic-gain-control 
oscillator — Hartley circuit. 



And in all tubes and transistors, the 
small-signal gain varies widely with the quiescent point when operating 
close to cutoff. 

By deriving the gain-control voltage from the oscillator output, any 
increase in the amplitude will create the conditions that will automatically 
reduce the gain and the signal. The response to a decrease in the output 
would be just the opposite. This action is similar to the thermally 
adjusted bridge previously discussed. To avoid distorting the individual 
cycle, the sinusoidal output is rectified and applied through a relatively 
long time constant to the gain-control element. Therefore the amplitude- 
defining equation may be written 

A(E)h = 1 

where E is the d-c control signal. 

Consider the general problem posed in Fig. 14-30, where the control 
terminal is indicated by 7. The output of the oscillator is coupled 
through the extra coil winding, rectified, and applied to this terminal. 
As shown, the gain is assumed to decrease with any increase in E y (or 
with an increase in the oscillation amplitude). For example, the linear 
approximation of the transconductance may be expressed as 

g n = <M - KE y (14-38) 



470 oscillations [Chap. 14 

where K is a positive constant. If g n were to increase with E y , then 
the control loop would be regenerative instead of degenerative and the 
output amplitude would increase until the active element limits. 

At any particular amplitude, the threshold of oscillation, as given by 
Eq. (14-23), is 

Li + M 
Taking the various coupling factors and amplitude transformations into 




<*) (b) 

Fig. 14-31. Two amplitude-controlled oscillators, (a) Junction tetrode transistor 
circuit — Hartley oscillator. The control current is injected into the second base 
(0 = 0o — Kit,). (6) A pentode tuned-plate oscillator where the suppressor is used 
to control the effective transconductance. 

account, this condition may be expressed as 



gmffp — K'Eq = 



U + M 
U + M 



E is the peak value of the sinusoidal output and will be found from the 
solution of this equation. 

Figure 14-31 shows two oscillators using automatic gain control to 
maintain the threshold operation. The first one is a transistor oscillator 
where the gain-control current is injected into the second base connection. 
Variations in over a range of 20:1 have been measured for control- 
current variations between and 2 ma. In the second circuit, the control 
signal is fed to the suppressor. An alternative arrangement, where the 
roles of the suppressor and the control grid are interchanged, has also 
been used; the greater effectiveness of the control grid in varying the 
gain of the tube would further improve the amplitude stability. 

In all the amplitude-controlled circuits, the variation of the con- 
trolled source with respect to the variable bias is difficult to express in 



Sec?. 14-7] almost sinusoidal oscillations 471 

general terms and will have to be evaluated in each individual case. 
Because the range of operation is severely restricted, a few measurements 
are usually sufficient. The best region in which to operate is the one 
where the variation of the controlling element with voltage is most pro- 
nounced. Even though satisfactory limiting has been obtained by this 
method, the selectivity multiplication of the bridge makes it superior 
when the output amplitude must be maintained constant. For an 
extremely high degree of stability, both methods of limiting may be 
incorporated in a single circuit. 



g ml - 3,000 /anhos 
^^-2,000/imhos 



Eplm 




E.l~ 

Fig. 14-32. Amplitude stability illustrated by describing functions. 

14-7. Amplitude Stability. As a second figure of merit for the oscil- 
lator, we might define a normalized amplitude stability factor 



„ x _ dx/x 
°* ~ dE/E 



(14-39) 



where x is the variable having the most pronounced effect on the ampli- 
tude. We interpret S% as meaning that a 10 per cent change in x pro- 
duces only a 10/S% per cent change in the amplitude of oscillation. 

In those circuits where the limiting depends on the nonlinearity of the 
active elements, S% may be found graphically by plotting a family of 
describing functions for various values of x. This process is illustrated 
in Fig. 14-32, where the variable of interest is the g m of the tube. At the 
intersection with the 1/00 line, the stability factor would be given by 



Oft. r^ Agm/ffm.T _ (gml 



gm2)/(gwl + gml) 



AE/E„ (E x - E 2 )/(Ex + E t ) 
For the values shown in Fig. 14-32, 

SJp = 3.8 
If the amount of feedback were to be increased, then the stable point of 



472 oscillations [Chap. 14 

operation would fall on the more nearly horizontal portion of the describ- 
ing function and the stability would improve. However, under these 
conditions, the harmonic content of the output would also increase, and 
as this represents a loading on the frequency-selective network, the 
frequency stability would be adversely affected. 

In the automatically stabilized circuits, such as the Wien-bridge 
oscillator of Fig. 14-27, any change in the base gain is compensated for 
by controlling the degree of bridge unbalance. Since A is always equal 
to d, the stability factor of interest is S S E . For the Wien bridge 

Ro + KE 11 



R a + KE + Ri 3 5 
By taking the differential, there results 



(14-40) 



(J?i + flo + K-E) 2 S 2 

But from Eq. (14-40), 



RlK - 2 dE = ±d6 (14-41) 



Rl 2 +\ (14-42) 



Ri + Ro + KE 3 6 

Since S » 1, the result of substituting Eq. (14-42) into Eq. (14-41) and 
of solving for the stability factor is 

S ^I & R^TKE VW 

As the rate of the resistance variation with voltage K increases, S% 
will approach 28/3 as an upper limit. Thus the amplitude stability is 
directly proportional to the amplifier gain and to the degree of bridge 
unbalance. Maximizing Eq. (14-43) by using a high-gain base amplifier 
also optimizes the frequency stability [Eq. (14-30)] and the harmonic 
rejection ratio [Eq. (14-16)]. 

For the particular values used in Example 14-2, S = 300, the amplitude 
stabilization becomes 

S E = 35.3 

This is many times the stability that could be obtained in the pure-resist- 
ance bridge unless the signal is distorted beyond recognition by the 
limiting in the tube. 

The automatic-gain-control circuit (Fig. 14-30) has a stability factor 
directly proportional to the rate of change of g m with the control voltage. 
Since 

gm =* £Uo — K'Ey 

ym 

For optimum stability a tube whose element has the most pronounced 
control over the transconductance should be chosen. Moreover, if the 



ALMOST SINUSOIDAL OSCILLATIONS 



473 



signal were to be amplified before applying it for control, the stability 
factor could be increased to almost any desired value. 



PROBLEMS 

14-1. (a) Find the frequency and necessary gain for sustained oscillation if the 
phase-shift network of Fig. 14-5 consists bf four equal RC sections instead of three. 

(6) Evaluate \H*\ and \Ki\ for this network. Compare the answers with the 
harmonic coefficients found for the three-section network. 

(c) Can the conditions for oscillation be satisfied with a two-section network? 
Explain your answer. 

14-2. (a) The network of Fig. 14-4c is used in the phase-shift circuit. Calculate 
the necessary amplifier gain, the frequency of oscillation, and the two harmonic factors. 

(6) Repeat part o if R and L are interchanged. 

14-3. The frequency of the phase-shift oscillator of Fig. 14-5 is to be controlled by 
varying only one of the three series capacitors over the range 0.1C < Co < IOC In 
which section should this capacity be placed if the range of frequency variation is to 
be maximized? What are the two bounding values of to, expressed in terms of 

o>, = 1/(V6«0? 

14-4. Calculate the input impedance of the three-/SC-section phase-shift network 
at «o- Under what conditions will it be invariant as the null frequency is changed? 
What is the oscillation frequency when the output impedance of the amplifier is equal 
to ij? What is the new value of the threshold gain? 

14-5. Verify Eqs. (14-15) and (14-17). Find H„ and K n for the Wien bridge for 
2 < n < 5. (Let S = 300.) 

14-6. The base amplifier of Fig. 14-7 has an unloaded gain of 200 and an output 
impedance of 5,000 ohms. It employs, for the input stage, a tube having y. = 20, 
r„ = 10 K, and a plate load of 5 K. If wo = 10* and fls = 10 K, specify the remain- 
ing parameters of the bridge. What is the actual value of 5? Make all reasonable 
approximations in the course of your solution and take into account the loading of the 
amplifier by the bridge and the loading of the bridge by the amplifier. 

14-7. Repeat Prob. 14-6 when the bridge is decoupled from the output stage of the 
base amplifier by means of a cathode follower. Its output impedance should be taken 
as 400 ohms. 

14-8. Calculate the oscillation frequency of the circuit of Fig. 14-13 when Rl 2> R- 
What are the harmonic-rejection factors? Plot the locus of the system's poles as A c 
increases from zero. 

14-9. In the circuit of Fig. 14-33, what is the minimum value of for sustained 
oscillations? At what frequency does this circuit operate? 



?+30v 




Fig. 14-33 



474 



OSCILLATIONS 



[Chap. 14 



14-10. A vacuum-tube tuned-plate oscillator (Fig. 14-16d) has a plate tank circuit 
consisting of a 10-mh coil and 100-/x/af capacitor. The grid coil is 5 mh, and the mutual 
inductance is only 1 mh. 

(o) Specify the frequency and the minimum value of n for oscillations. 

(6) If the plate tank circuit has Qo = 50 and if r p «• 5 K, how would the frequency 
change? Under this condition, what is the minimum value of g m needed for proper 
operation? 

14-11. Consider the following two cases of the general feedback problem posed in 
Fig. 14-15a. All elements are assumed to be purely reactive. 

(a) When there is no mutual coupling between Xi and Xj. 

(6) When there is mutual coupling but Zs is an open circuit. 

Under these conditions, calculate separately the gain and the feedback factor and 
solve for the frequency of oscillation. If a high-/* tube is used, what simplifications 
may be made in evaluating /3 for the Colpitts and for the tuned-plate oscillator? 

14-12. A tube having n = 20 and r, = 10 K is used in a Hartley oscillator where 
the two coils are completely isolated (M =0). It is to be tuned over a range of 500 
to 1,500 kc by using a capacitor that varies from 300 nid down to 30 nnl . If the Q of 
the complete coil is 20 at 1 megacycle, specify the two inductances that will permit 
oscillation over the complete range of frequencies. Be svu-e to check the end points 
and to account for the loading effect of the tank resistance. 

14-13. Verify Eqs. (14-25) and (14-27) and the frequency and self-starting condi- 
tions given in the text for the Hartley and Colpitts transistor oscillators. 

14-14. A transistor used for a Colpitts oscillator has r n = 500, r c — 1 megohm, 
and f) = 50. The nominal frequency of oscillation is too = 10'. Plot the deviation 
from wo as a function of the tank impedance. Assume Qo remains constant. Take a 
range of tank impedances from 1 K to 1 megohm. Under what conditions will the 
frequency of oscillation be least susceptible to changes in the external loading? 

14-16. Figure 14-34 shows a three-phase transistor oscillator where the three 
secondaries are symmetrically loaded. 

(a) Calculate the frequency of oscillation, assuming threshold operation. 

(b) What is the maximum load that can be applied at the secondary without caus- 
ing the cessation of oscillation ? 

(c) Calculate the frequency-stability factor and the harmonic-rejection factor \H t \. 




(3=50 
r{ i-500 
r c -lM 
»-10:l 



Fig. 14-34 



ALMOST SINUSOIDAL OSCILLATIONS 475 

14-16. Compare the frequency-stability factor of the phase-shift oscillator of Figs. 
14-5 and 14-10 when the frequency-determining network consists of three sections to 
one where four RC sections are used. 

14-17. (a) Verify the frequency-stability factor for the Meacham bridge. 

(6) Two identical crystals are used in this bridge instead of one. The second one 
replaces flj in Fig. 14-22. If each has Q = 10 6 and the degree of bridge unbalance is 
a = 100, what is the new frequency-stability factor? The nominal resonant fre- 
quency of the crystal is 10' cps, and the change in the amplifier phase shift may reach 
a maximum of 10°; what is the effective change in frequency? 

14-18. Find the frequency-stability factor for the transistor current-controlled 
oscillator of Fig. 14-33. 

14-19. A Hartley oscillator is adjusted for threshold operation at/o = 1 megacycle, 
using C = 250 jipf. As the g m of the active element changes, the tap on the coil is 
accordingly changed to maintain the oscillations. Assume that there is 10 /»/if of 
stray capacity from the input to the output of the active element; we wish to mini- 
mize the effect of the stray reactance on the frequency of oscillation. Should a 
device with a low or high value of transconductance be used? Explain your answer 
and justify it fully by comparing two circuits, one which uses a tube having n = 100 
and r, = 50 K and the other a transistor where r' u = 500 ohms, /3 = 100, and n «- 
100 K. Which device would make a more stable oscillator? 

14-20. A transistor used in a Colpitts oscillator has the following parameters: 
r u = 20, r c = 2 megohms, and a = 0.98. The oscillator uses a 50-mh coil to set the 
nominal frequency of u — 5 X 10" radians/sec. We can expect stray capacity of 
5 ju/*f between the base and collector connections. 

(a) If the circuit is initially adjusted so that the loop gain is twice the threshold 
value, by how much will the frequency shift as a result of the stray capacity? 

(6) Repeat part o for a transistor having a = 0.99. The nominal gain is again 
twice the threshold value. 

(c) If the tank has a loaded Q = 20 and if the phase of the base amplifier is 10°, by 
how many cycles would the frequency shift from w ? Compare the answer with that 
obtained in parts a and b. 

14-21. The tuned circuit in the Colpitts oscillator of Prob. 14-20 has a Q of 20. 
Without accounting for the effects of the stray capacity, calculate the reactance 
which would have to be inserted into the base lead in order to return the frequency of 
oscillation to wo. Repeat the calculations for a stabilizing reactance placed in series 
with the emitter. 

14-22. Would it be feasible to minimize the effect of stray capacity by means of a 
stabilizing reactance? Explain and justify your answer by considering the vacuum- 
tube Hartley circuit of Fig. 14-16b. 

14-23. Given the Colpitts oscillator of Fig. 14-23 adjusted to 25 per cent above 
the threshold value of gain. The parameters of interest are Ebb = 150 volts, r, — 
20 K, /i = 50. The tank circuit has a loaded Q = 20 and an input impedance of 30 K 
at its resonant frequency of 10 6 cps. Bt is chosen to give a symmetrical transfer 
characteristic. 

Verify Eqs. (14-35) and (14-36). 

Plot the describing function by first finding the maximum value of the fundamental 
component of plate voltage in the linear region; next the output when the cosinusoidal 
signal is clipped from —15 to +15°; and then when it is clipped from —30 to +30°; 
etc. Represent the describing function by a sequence of straight lines connecting 
these points. Find the approximate amplitude of oscillation. Calculate <SJ at the 
operating point. 



476 oscillations [Chap. 14 

14-24. Repeat Prob. 14-23 when the limiting is controlled by the assumed perfect 
grid circuit clamping. 

14-26. A transistor tuned-collector, tuned-base oscillator, operating at 10 6 cps, has 
an effective collector-circuit tank impedance of 5 K. Em, = 20 volts, r' u = 100, and 
= 100. The circuit is adjusted to 50 per cent beyond the threshold. The bias is 
derived by assumed ideal base-circuit clamping. 

(a) Approximate and plot the describing function and find the amplitude of 
oscillation. 

(6) Evaluate Sj,™ at the operating point. 

14-26. Design an amplitude-regulated Meacham-b ridge transistor oscillator using 
the 6.3-volt pilot lamp of Fig. 14-28 as the control element. The peak amplitude 
across the bridge should be 3 volts. The crystal, at its series-resonant frequency, 
may be replaced by a 20-ohm resistor. Specify all the elements of the bridge and 
draw the basic amplifier indicating the various elements used for impedance matching. 

14-27. Find the output amplitude of a Wien-bridge oscillator where the bridge is 
initially unbalanced by a factor $ = 100 and where the amplitude-control element is 
a thermistor having the approximate characteristics R = 10,000 — 25QE ohms. The 
associated pure-resistance bridge arm is 3,000 ohms. 

14-28. Repeat Prob. 14-27 if the remaining resistance arm is replaced by a tungsten 
lamp having the approximate characteristics R' = 1,000 4- 100E. Thus two tem- 
perature-dependent elements are used in the bridge. Evaluate S S E . 

14-29. Evaluate the amplitude-stability factor of the Meacham bridge of Prob. 
14-26. What combination of arm resistances would maximize this factor? (The 
bulb may be changed if necessary.) 



BIBLIOGRAPHY 

Anderson, F. B.: Seven-league Oscillator, Proc. IRE, vol. 39, no. 8, pp. 881-890, 1951. 
Bode, H. W.: "Network Analysis and Feedback Amplifier Design," D. Van Nostrand 

Company, Inc., Princeton, N.J., 1945. 
Bollman, J. EL, and J. G. Kreer, Jr. : Application of Thermistors to Control Networks, 

Proc. IRE, vol. 38, no. 1, pp. 20-26, 1950. 
Bothwell, F. E. : Nyquist Diagrams and the Routh-Hurwitz Stability Criterion, Proc. 

IRE, vol. 38, no. 11, pp. 1345-1348, 1950. 
Chance, B., et al.: "Waveforms," Massachusetts Institute of Technology Radiation 

Laboratory Series, vol. 19, McGraw-Hill Book Company, Inc., New York, 1949. 
Edson, W. A.: "Vacuum-tube Oscillators," John Wiley & Sons, Inc., New York, 

1953. 
Ginzton, E. L., and L. M. Hollingsworth: Phase-shift Oscillators, Proc. IRE, vol. 29, 

no. 1, pp. 43-49, 1941; also corrections, vol. 32, no. 10, p. 641, 1944. 
Hooper, D. E., and A. E. Jackets: Current Derived Resistance-Capacitance Oscil- 
lators Using Junction Transistors, Electronic Eng., vol. 28, pp. 333-337, August, 

1956. 
Keonjian, E. : Variable Frequency Transistor Oscillators, Elec. Eng., vol. 74, no. 8, 

pp. 672-675, 1955. 
Llewellyn, F. B. : Constant-frequency Oscillators, Proc. IRE, vol. 19, no. 12, pp. 2063- 

2094, 1931. 
Meacham, L. A.: The Bridge Stabilized Oscillator, Bell System Tech. J., vol. 17, 

pp. 574-590, 1938; also Proc. IRE, vol. 26, no. 10, pp. 1278-1294, 1938. 
Nyquist, H.: Regeneration Theory, Bell System Tech. J., vol. 11, pp. 126-147, 1932. 



CHAPTER 15 
NEGATIVE-RESISTANCE OSCILLATORS 



The energy needed to sustain oscillations can be supplied by shunting 
the dissipative tuned network with the negative driving-point resistance 
of an active element. The solution of this problem is well defined, and 
from it we can draw some general conclusions as to the behavior of even 
those circuits where it is difficult to isolate the two terminals across 
which the negative impedance is developed. This chapter serves not 
only as an extension of the previous techniques, but also as an introduc- 
tion to the difficult problems posed by nonlinear differential equations. 





(b) 

Fig. 15-1. (a) Basic negative-resistance oscillator; (6) typical volt-ampere character- 
istic with superimposed conductive load line. 

15-1. Basic Circuit Considerations. The response of the voltage- 
controlled circuit of Fig. 15-la, with which we are initially concerned, is 
defined by the single node equation 



dV 



U 



V dt + In = 



(15-1) 



where the bottom of the GLC circuit is taken as the reference node. Since 

V = v a - V„ and /„ = /„ + in(t) 

by differentiating Eq. (15-1) and collecting terms, the time-varying 
response will be given by the solution of 



d 2 v„ 1 ( r dv* din\ , 

d< 2 + C\ drrfij 1 " 



LC 







(15-2) 



477 



478 oscillations [Chap. 15 

For small-signal operation, 

di„ _ din dvn _ dt)n 
dt ~ dv„ dt ~ d dt 

where g is the incremental conductance of the VNLR measured at the Q 
point of the volt-ampere characteristic (point Q in Fig. 15-16). With the 
above substitution, the basic incremental defining equation may finally be 
written as a quasi-linear differential equation: 

On + ^ (G + g)v n + -L v n = (15-3) 

The linearization of the differential equation, which results from the 
implied assumption of a constant value of g, will not be valid once the 
signal amplitude carries the operating path into the dissipative regions of 
the volt-ampere characteristics. To find the amplitude of oscillation, the 
nonlinear differential equation of the system must be solved. Before 
treating this more difficult problem, we shall find the necessary and 
sufficient conditions for the circuit of Fig. 15-1 to sustain almost sinus- 
oidal oscillations from the quasi-linear equation. 

Equation (15-3) has two roots, which are located at 



Pw =~^(G + g)±j 4lC-W^ G + g) * (15-4) 

= a ± ja 

These roots determine the nature of the time-varying response of the 
negative-resistance oscillator, and therefore we shall investigate their 
location as a function of g. In order to have a growing transient, a > 0, 
which corresponds to 

G < -g (15-5) 

Equation (15-5) can be satisfied only when the conductive load intersects 
both the dissipative and the negative conductance portion of the charac- 
teristic. If there is only a single intersection on the negative portion of 
the curve, the response will decay instead of increasing with time. 

The second restriction which must be imposed on the roots, if the 
response is to be oscillatory, is that u must be real. From Eq. (15-4), 

AC 

■£>(& + gY (15-6) 

The boundaries delineating the inequality of Eq. (15-6) are two straight 
lines, which satisfy 



-G + 



&*>* 



Sec. 15-1] 



NEGATIVE-RESISTANCE OSCILLATOKS 



479 



These regions are shown in Fig. 15-2. We are interested in only that 
portion of the plane below the g = —G line, because it is here that a self- 
excited signal is generated, and we are primarily concerned with the 
narrow segment denning the sinusoidal oscillatory response. 




Fig. 15-2. Modes of operation of the circuit of Fig. 15-1 as a function of the external 
and internal conductances. 

At the threshold of sustained oscillations, i.e., when G = —g, the 
external conductance is tangent, at the Q point, to the volt-ampere char- 
acteristic and the negative conductance of the active element exactly 
cancels the positive conductance of the tuned circuit. The purely 
imaginary poles are located at 

Pij = ±j ^m 

For any other value of the external conductance lying in the oscillatory 
zone, the load line also intersects the positive-resistance regions, and as 
the oscillations build up, these segments will limit the amplitude to a 
finite value. 

When G is small compared with the magnitude of the negative con- 
ductance, u becomes imaginary and the circuit operates as an astable 
multivibrator. In the relaxation region, the response is similar to that 
discussed in Chap. 11, where the second energy-storage element appeared 
as part of the driving-point impedance of the active element instead of 
as part of the external load. To return the operating point to the 
sinusoidal oscillation region, we can load down the tuned circuit and 
hence reduce its effective Q. 

We might further note that as y/L/C is increased, the width of the 
oscillatory region shrinks. A circuit operating properly under load 
may switch into the relaxation zone once the load is removed. Since 



480 



OSCILLATIONS 



[Chap. 15 




the design of the tank controls the width of the oscillatory region, by 
minimizing -\/L/C, the performance of the oscillator becomes much less 
critical. 

The possible modes of system operation may also be explained by 
following the migration of the system's poles as G is reduced toward 

zero (as along the dashed line of 
Fig. 15-2). Initially, the circuit is 
dissipative and the two poles lie on 
the real axis (Fig. 15-3). As G 
decreases, the poles first coalesce 
and then separate along the arcs 
of a circle. This occurs at point 
A in Figs. 15-2 and 15-3. When 
G = —g, they lie on the imaginary 
axis. A further reduction in G 
permits an exponentially increasing 
output, with the poles moving into 
the right half plane; the negative 
conductance now predominates. Eventually they again coalesce (point 
C in Fig. 15-2) and separate along the positive real axis. Here the circuit 
operates as a relaxation oscillator. Consider also the plot of the roots of 
the negative-resistance switching circuit presented in Fig. 11-26. 

Example 16-1. To place some of the thoughts to be presented in this chapter in 
their proper perspective, we shall now represent the tuned-plate oscillator, which was 
treated from the feedback viewpoint in Chap. 14, as a negative-resistance circuit. 
Consider Fig. 15-4a when, in the active region, the grid winding is unloaded. With 
the proper polarity connections of the feedback coil, the drive voltage becomes 

M 
e g = _ e p 

Substituting into the controlled source of Fig. 15-4&, we see that the current flow is 
proportional to the voltage developed across the two terminals. Hence the con- 
trolled source may be replaced by the negative conductance 



Fig. 15-3. Pole migration along the 
dashed path of Fig. 15-2 with decreasing 
values of G. Points of interest corre- 
spond to those labeled in Fig. 15-2. 



M 



gm 



leading to the equivalent model of Fig. 15-4c. 

For sinusoidal oscillations, this negative conductance must lie within the appro- 
priate region of Fig. 15-2. By lumping r p together with the external conductance, 
the restriction on the transconductance is given by 



L + G + Jf > M^ > L + G 

r p ' L L r p 



Figure 15-4d illustrates how the negative-resistance region II is bounded by the dis- 
sipative segments; region I is where the grid conduction limits the amplification; and 



Sec. 15-1] 



NEGATIVE-RESISTANCE OSCILLATOES 



481 



region III is where the large-signal amplitude drives the tube into cutoff. Such 
regions and their bounds are readily found from the piecewise-linear model. 




T St 1 - &m e g 




(b) 




(c) (d) 

Fig. 15-4. (a) Tuned-plate oscillator; (6) incremental-plate circuit model; (c) equiva- 
lent negative-resistance circuit; (d) volt-ampere characteristics as drawn through the 
quiescent point of operation. 

CNLR Oscillator. If a CNLR is employed as the energy source in a 
negative-resistance oscillator, then the normal mode of operation is 
satisfied by an external series-resonant circuit (Fig. 15-5). Since this 
is the dual of Fig. 15-1, the single defining 
equation is written on a loop basis : 



i£ + a/ + 



H 



I dt + Vn = 



After differentiating and collecting terms, 
we obtain 



din 



+ 




Fig. 15-5. CNLR oscillator con- 
figuration. 



W ' L V du) dt ^ LC 

(15-7) 

which is the dual of Eq. (15-2). All arguments employed with respect to 
the voltage solution of Eq. (15-2) are immediately applicable to the cur- 
rent solution of Eq. (15-7). 

For example, the roots of the quasi-linear differential equation, where 
we define 

— = r 
din 



482 oscillations [Chap. 15 

are now located at 



And the new boundary conditions are denned by the equations 
For a > 0: R < -r 

For c real: -R + J^ > r > -R - J~ 

These should be compared with the conditions previously derived for the 
VNLR circuit and with the plot of Fig. 15-2. 

15-2. First-order Solution for Frequency and Amplitude. In order to 
keep the nonlinearity in the forefront, Eq. (15-2), defining the complete 
response of the basic voltage-controlled negative-resistance oscillator of 
Fig. 15-1, may be rewritten 

S » + ?( 1+ gct)^ + zi W »= < 15 " 8 > 

This is in the form of the Van der Pol nonlinear differential equation 

v + ef(v)i> + coo 2 w = 

For almost sinusoidal oscillations, ef(v) must remain small over the 
range of interest. When it is zero, the Van der Pol equation reduces to 
the equation of simple harmonic motion, 

v + coo 2 v = 

which has the solution 

v = A cos (o>o< + 4>) 

If the nonlinear term is large, then Eq. (15-8) can be easily solved only 
for some specific analytic functions of tf(y). 

To find the frequency and amplitude to a first approximation, we shall 
apply the method of equivalent linearization. We shall assume that 
over any single cycle the gross behavior of the system may be represented 
by an equivalent linear circuit, and we shall neglect any variations taking 
place during the individual cycle. 

Choosing as a suitable trial solution for Eq. (15-8) 

v„ = E(t) cos at 
and substituting into Eq. (15-8) and collecting terms yields 
[d*E „_ , E , G(, , ldi n \dE~[ 



Sec. 15-2] negative-besistance oscillators 483 

For almost sinusoidal oscillations we may further assume that E(t) 
remains essentially constant over any single cycle of the steady-state 
response. Since 

Eq. (15-9) reduces to 

(-"• + eg) cos * - it i 1 + h £) 8in * - ° (15 - 10) - 

To evaluate the average response, Eq. (15-10) is now multiplied by 
cos u>t and integrated over a complete cycle: 

jT (-" 2 + w) cos2 ui d ^ - jT it( 1 + g a) sin at cos -* d( - at) 

= 

If the circuit is adjusted close to the threshold of oscillation (to mini- 
mize the distortion of the sinusoid) , then the degree of limiting, and hence 
the nonlinearity, is small. The coefficient of sin wt cos wt in the second 
integral is this very nonlinearity. It plays a major role in determining 
the amplitude, but since it is small, it will have a relatively minor effect 
on the frequency. By assuming that this coefficient is essentially 
constant over the period, the second definite integral would be identically 
equal to zero. Consequently, the approximate frequency of oscillation, 
given by the vanishing of the coefficient in the first integral, is the natural 
frequency of the tuned circuit 

1 



0)0 = 



Vlc 



We shall now investigate E(f) by rewriting the fundamental circuit 
equation. With the assumed cosinusoidal solution, Eq. (15-2) becomes 

din \ n d 2 E , n dE,(\ 2 „\ _"| 
- df = [ C W +G Tt + (l - ^E^coswt 

- UwC ^ + wGe\ sin wt (15-11) 

By again assuming that all coefficients are approximately constant over 
the cycle, the first expression on the right-hand side of Eq. (15-11) may 
be eliminated by multiplying through by sin wt and integrating over a 
complete cycle. The equation of interest reduces to 

-w f 2 ' — sin wtd(wt) = - (2<oC^ + <&E\ P'sin 2 wtd(wt) 

(15-12) 



484 Oscillations [Chap. 15 

The left-hand side of Eq. (15-12) may now be integrated by parts: 



Jo d(o>i 



A sin oit d{(d) = — u(i n sin ut)\ + co 
>t) |o 



/•2x 

/ in cos ait d(wt) 
Jo 

= 01 J i n cos (at d(oit) (15-13) 
Jo 



But the right-hand side of Eq. (15-13) is proportional to the peak value 
of the fundamental component of the current flow into the VNLR. Thus 
the solution of Eq. (15-12) is 



omI\ m 



Kt? 



or 



~~dt~2C (7lm + GE) 



+ oiGE W 



(15-14) 



(15-15) 



At equilibrium the oscillations are constant, dE/dt = 0, and 

Ilm = —GE 

The amplitude satisfying Eq. (15-15) is found by plotting a describing 
function, i.e., the fundamental component of the current into the VNLR 
vs. the applied-voltage excitation, and solving for the intersection 
with the conductance line. Two typical curves appear in Fig. 15-6. 




Fig. 15-6. Two describing functions for the basic nonlinear oscillator of Fig. 16-1: 
curve 1 for a circuit biased in the center of the negative-conductance region, and curve 
2 for a circuit biased in the positive-resistance region near the peak point. 

Curve 1 appears when the circuit is normally biased within the negative- 
conductance region. For small signals, the current and voltage are out 
of phase. As the amplitude of oscillation increases, the operating point 
is driven into the dissipative regions for a portion of the cycle and the 
fundamental current begins decreasing. The second curve corresponds 
to a device normally biased within the dissipative region, such as at point 
A or B in Fig. 15-1. For small signals, the driving-point impedance is 
positive and the current is in phase with the applied excitation. A larger 
amplitude, where the average conductance over a cycle is negative, per- 
mits sustained oscillations. 

In order to see which of the multiple intersections represent stable 
points of operation, we return to Eq. (15-14) and consider the effects of 
a small variation in E about the amplitude satisfying Eq. (15-15). 



Sec. 15-2] negative-resistance oscillators 485 

w = ~"k [Ie + AI + G{E ' + AE)] 

and I. and E. are the values at the equilibrium point. From Eq. (15-14) 
we know that 7, + GE, = 0. Consequently 

^ m _ifM+flW (15-16) 

dt 2C\AE^ J 

For stability, any change in E must establish the conditions that will force 
a return to the equilibrium point. It follows that if the initial perturba- 
tion AE is positive and momentarily increases the output, dE/dt must 
be negative in order for the voltage to decrease with time back to the 
stable amplitude. This is possible only when the term in the parentheses 
of Eq. (15-16) is positive. The condition for stability becomes 

f E + G>0 (15-17) 

Equation (15-17) says that the total incremental conductance at the 
stable point must be positive, or in other words, the circuit must appear 
dissipative with respect to small variations about the equilibrium point. 

In order for any oscillator to be self-starting, the origin must be a 
point of unstable equilibrium. The oscillator of curve 1 will build up to 
point a, the single stable point. However, if the quiescent point is along 
the positive-resistance segment, then the origin will be stable and the 
circuit cannot start by itself (curve 2). Once the circuit is externally 
excited to, or past, the unstable point b, the build-up will continue to the 
second point of stable equilibrium, point /. 

To evaluate the describing function analytically, the characteristic of 
the VNLR may be approximated by a power series which is written with 
respect to the quiescent point. 

in = a,u„ + awn* + a s w„ 8 + • • • (15-18) 

where the various coefficients are found from the curve. Because of the 
nature of the external circuit (which is a relatively high-Q tank), only the 
fundamental component of voltage may be developed across the terminals 
of the nonlinear element. Furthermore, when the nonlinearity is small, 
the fundamental is the only current component of interest. 

Substituting v n = E x cos ut into Eq. (15-18) and collecting the coeffi- 
cients of the cos utt terms results in 

h = oj^! + HazES + HasES + ■ ■ ■ (15-19) 

For each value of the assumed voltage excitation, the current response 
term is now known. Cross plotting gives the describing function shown 
in Fig. 15-6. 



486 



OSCILLATIONS 



[Chap. 15 



The stable amplitude may also be found by realizing that at the thresh- 
old of oscillation, the negative conductance of the active element exactly 
cancels the damping of the tuned circuit. Again, only the fundamental 
components are of interest; at the harmonics the current and voltage are 
in quadrature and do not represent power supplied to the tank. From 
Eq. (15-19), the average conductance of the VNLR over a cycle is 



Gn = W 1 = <Xl + l a * El * + I atEii + 



(15-20) 



Equation (15-20) may be plotted as shown in Fig. 15-7 (the two curves 
drawn correspond to the two current-describing functions of Fig. 15-6). 




Fig. 15-7. Two possible conductance describing functions. Curve 1 holds when the 
quiescent point is at the center of the negative-conductance region; curve 2, when the 
quiescent point is in the positive region near the knee of the curve. These correspond 
to the current describing functions of Fig. 15-6. 

By superimposing the G line on the negative conductance-describing 
function, the amplitude satisfying G n + G = is finally found. 

Example 16-2. A device having the almost symmetrical piecewise-linear character- 
istics of Fig. 15-8 is used in the basic oscillator circuit of Fig. 15-1. It is initially 
biased in the center of the negative-resistance region, at Qi, where I c = 40 ma and 
V„ = 20 volts. 

By taking three coordinates on the curve, measured with respect to the Q point, we 
can construct a cubic that will adequately approximate the characteristic. Choosing 
the following points: ( — 10 volts, ma), (—5 volts, 20 ma), and (+5 volts, —20 
ma), the approximating polynomial becomes 

H = -5.33» + 0.0533«« ma 

Thus the conductance describing function, as given by Eq. (15-20), is 

Oni = -5.33 + 0.04B,' millimhos 

and it is plotted in Fig. 15-86. When, for example, the conductance of the tank is 
3.5 millimhos, the circuit will oscillate with a peak amplitude of 6.75 volts. 

Suppose that the quiescent point is shifted to Qi, where V„ = 17 volts; by how much 
would the amplitude change? Rather than solve for a new describing polynomial, 
for the purposes of this discussion we can simply shift the previously derived curve to 
correspond to the new origin. The shifted curve is denned by 

ij 5.33(f - 3) + 0.0533(» - 3)» + A 



Sec. 15-3] 



NEGATIVE-RESISTANCE OSCILLATORS 



487 



where the constant A must be included to make i - when v = 0. This equation 
reduces to 

U « -3.89» - 0.48»» + 0.0533»« ma 
Consequently, the new conductive describing function is characterized by 

Gni = -3.89 + 0.04JS, 2 millimhos 
and the new peak amplitude stabilizes at 3.5 volts. 



60 

ra 40 

" 20 


. 


























\e 2 




















Qi 







































> 1 


1 


5 2 


25 30 35 40 45 V 
volts 






(a) 

2 4 


i 


i 


10 




1.0 
* 2.0 
G„ 3.0 














Ex 




















G n y 
























4.0 


















^/& 


iti 








5.0 
















6.0 




W 













Fig. 15-8. (a) Volt-ampere characteristic; (6) the conductance describing functions for 
Example 15-2. 

15-3. Frequency of Oscillation to a Second Approximation. First 
Method. In order to obtain a more accurate expression as to the fre- 
quency of oscillation, we shall now solve the Van der Pol equation with 
the simple assumption that the voltage waveshape is periodic. This 
equation is 

v + ef(v)v + co 2 w = 

where wo 2 = 1/LC and where 

for the general circuit of Fig. 15-1. As the first step toward the evalua- 



488 oscillations [Chap. 15 

tion of the oscillation frequency, the Van der Pol equation is multiplied 
by the unknown solution and integrated over the unknown period. 

f 2 ' wdut + e f 2 * f(v)vv dot + co 2 [ 2 *v i dust = (15-21) 

Let us now consider each term of Eq. (15-21) individually, starting 
with the one that contains the system's nonlinearity. 

F(») = t J** f(v)vv dut (15-22) 

By expanding the volt-ampere characteristic into a power series, 

* = OiW + ow 2 + a a v 3 + a 4 u 4 + • • • 
the nonlinearity becomes 

/(„) =(? + ^ = G + a! + 2a 2 w + 3a 3 t> 2 + • • ■ (15-23) 

The necessary assumption of a periodic solution leads to the expression 
of v and ii as Fourier series. 



v = V V k cos (host + 4> k ) (15-24a) 

and i, = - \ kaiVk sin (hat + 4>k) (15-246) 

Consequently vv consists solely of terms which are the product of two 
time-varying signals in quadrature, a sinusoid and a cosinusoid; it does 
not contain any d-c terms. The substitution of v into Eq. (15-23) and 
the collection of terms will result in a d-c component and a series of 
cosinusoidal terms, i.e., the fundamental and all harmonics. The 
integral of interest [Eq. (15-22)] reduces to the evaluation of various 
factors of the form 

/ * (dc + cos nut) (cos koit) (sin moit) did 

over a complete cycle of the fundamental where k, m, and n are nonzero 
positive integers. But such an integral is always equal to zero. Thus 
we have proved that 

F( v ) = e [ 2r f(v)vvd<j>t = 



Sec. 15-3] negative-resistance oscillators 489 

Returning to Eq. (15-21), the first integral may be evaluated by parts: 

I vv doit = u I j—.v doit = oi I vd(v) 

Jo Jo doit Jwt-o 

\U — 2ir fat = 2r 

oi v 2 dt (15-25) 

Jat-0 



\ut-0 



It follows directly from the argument employed with respect to the 
integral of Eq. (15-22) that 

\ut = 2t 

vv \ = 

M-0 

and as a result 

P* vv doit = - f*" t> 2 doit (15-26) 

The right-hand integral may be identified as the mean-square value 
of the derivative. Equation (15-21) finally reduces to 

f*' y 2 doit = oio 2 f 2 " v 1 doit (15-27) 

When the Fourier series of Eqs. (15-24a) and (15-246) are substituted 
into Eq. (15-27), only the squared terms would contribute to the final 
answer: all cross products integrated over a cycle equal zero. The 
evaluation of Eq. (15-27), consistent with the assumed periodicity of the 
solution, results in 



H 


V kWV k " 


= Vim 1 ^ V * 2 




t-1 


fc-1 

SO 

y v k * 




\"o/ 


» 
y kw k * 



- -_£=i (15-28) 



Equation (15-28) proves that when limiting introduces distortion 
terms, the frequency of oscillation is always depressed below the nominal 
resonant frequency of the tuned circuit, and that the greater the distor- 
tion, the greater the frequency depression. This equation is somewhat 
difficult to use in the form derived, since for small amounts of distortion, 
oi = wo. To simplify the expression, let us consider 



«o 2 



y w t(** -div 

k = l _ k^l 

00 CO 

^ k*V k * J k*V„* 



1_4=1- Jt=l = »=1_ (15-29) 



*-i *-i 



490 oscillations [Chap. 15 

However, 

1 _ w^ _ (too + o>)(o>o — to) ^ — 2 Ao> 

O>0 2 O)o 2 O)o 

where the approximation holds when o> is close to o>o and where the 
frequency deviation is Au s u - o> . Furthermore, by restricting the 
simplification of Eq. (15-29) to signals containing only slight distortion, 
the fundamental term greatly predominates. Because 



Fi 2 » £ kW k * 



k-2 



the denominator of Eq. (15-29) may be approximated by Vi 1 . 
The frequency deviation may finally be expressed in the form 



Aw --i£(fc 2 -l)£I (15-30) 



O) 

k-2 



For example, when the VNLR characteristics are symmetrical about the 
Q point, only odd-harmonic terms will be present in the output. If the 
principal one is the third harmonic and if it is even 5 per cent of the 
fundamental, 



>xsx(^)'-- 



^--jX8X[^l^| --1.0% 



Contributions of the higher-harmonic components will depress the fre- 
quency even further. 

Second Method. Our second method of finding the frequency depres- 
sion, because of the nonlinearity of the negative conductance character- 
istic, will be based on a power balance of the circuit. The ideal negative- 
resistance device, by its very nature, is unable to store any energy and is 
also unable to supply any reactive power to the external circuit. Like 
any characteristic that is a single-valued function of the independent 
variable, it does not exhibit hysteresis; consequently the line integral 
of i dv over a cycle is 

fi dv = (15-31) 

We shall therefore sum up the reactive power terms at each harmonic and 
set the result equal to zero. The particular frequency at which this 
condition is satisfied will be the frequency of oscillation. 

As before, the unknown periodic voltage may be expressed as a Fourier 
series. When it excites the tuned circuit of Fig. 15-la, the reactive 
component of the current at each harmonic will be given by 

I' h = h sin fe = B k V k (15-32) 



Sec. 15-3] 



NEGATTVE-BESISTANCE OSCILLATORS 



491 



where B k is the susceptance at the fcth harmonic. I' k is in quadrature 
with the voltage component of the same frequency. The trajectory 
traversed on the volt-ampere charac- 
teristic, due to this current, is an 
ellipse having the area 

A' k = *V k I' k = *V k I k sin <f> k 

This path is shown in Fig. 15-9. 

During one period of the funda- 
mental, the harmonic circumscribes 
this path A; times, tracing out the area 

A k = TkV k I k sin <t> k = *hV k *B k 

(15-33) 

But this is simply vk/2 times the reac- 
tive power. Since no energy is stored 
by the VNLR, Eq. (15-31) must be satisfied, 
expressed as 




V 

Fig. 15-9. Negative-resistance char- 
acteristic showing the path traversed 
by the harmonic components of volt- 
age and current. 

This condition may now be 



no » 

V A t = = tVJB x + r ^ kV* 2 B k 

t-l *-2 



or, in terms of the reactive power, 



Pis + y kP kB = 



(15-34) 



(15-35) 



At some sufficiently high harmonic, usually at and above the second, the 
tuned circuit appears capacitive. In order for Eq. (15-34) to hold, JSi 
must have a sign different from that associated with B 2 , B s , etc. This 
is possible only if the tank appears inductive at the fundamental. We 
have therefore proved once again that the frequency of oscillation must be 
below the resonant frequency of the tuned circuit. 

The susceptance of the tuned circuit may be expressed as 



B = Cm 



/ CO _ COo\ 

\oio u> ) 



(15-36) 



In the vicinity of resonance Eq. (15-36) may be approximated by 

Bi S 2C Ace 
and at the harmonics Eq. (15-36) reduces to 



B k ^ Uo C- 



1 



492 oscillations [Chap. 15 

Substituting these two terms into Eq. (15-34) and solving results in 

£«_ _ I V (k* - 1) ^ 

which is identical with Eq. (15-30). Furthermore, since from Eq. (15-34) 

IiVi sin </>i = — ZklhVk sin <j> k 
the depression in frequency is also given by 

where <t>i is the phase angle of the fundamental current with respect 
to the fundamental voltage. 

Synchronization. The automatic balancing of the circuit, which shifts 
the frequency to a point where the net reactive power is equal to zero, 
may be utilized for frequency entrainment or synchronization. Suppose 
that an external signal that is some rational ratio of the desired funda- 
mental is applied to the basic negative-resistance oscillator. Because 
the tank is primarily reactive at this frequency, the additional power 
injected (P,&) will be reactive and Eq. (15-35) will become 

Pi+ Y kP k + kP, k = (15-38) 

4-2 

Depending on the phase of the injected signal, the effective reactive power 
P,h can be either positive or negative; the frequency at which Eq. (15-38) 
is satisfied may be shifted in either direction. 

For example, if P, k is inductive and large enough to cancel the other 
harmonic terms, then the oscillation frequency could be returned to w . 
On the other hand, if P, k is capacitive, the net harmonic power is 
increased and the frequency would be further depressed. Essentially, 
the injection of this additional power changes the effective susceptance 
of the circuit. 

15-4. Introduction to Topological Methods. Nonlinear systems which 
are described by a differential equation of the second order may also be 
studied with the air of various topological constructions. This form of 
representation is important because it enables us to gain perspective as 
to the totality of the possible behavior of the system without having to 
solve the actual differential equation for the time response. The periodic 
solution will appear as a closed contour on some graph. And as the 
graphically presented nonlinearity will be used directly — without con- 
version to an approximating polynomial — grossly nonlinear as well as 



Sec. 15-4] 



NEGATIVE-RESISTANCE OSCILLATORS 



493 



quasi-linear circuits may be treated. This is not to say that the topo- 
logical solutions do not have their own limitations; they do. In the 
course of the following arguments some of the drawbacks will become 
apparent. 

As a convenient starting point, consider the second-order linear differ- 
ential equation 



x + bx + <ao 2 x = 



(15-39) 



Phase plane 



Two initial conditions, which give information as to the initial energy 
in the system, are necessary for the complete solution; usually the 
position and velocity at t = are specified. 

For the simplest case of Eq. 
(15-39), we set 6 = 0, which re- 
duces the problem to one of simple 
harmonic motion. The well-known 
solution for x in terms of time is 

x = K cos (adt + <£) (15-40) 

But in order to evaluate the two 
constants, we usually also find the 
derivative. 




Concentric 
ellipses 



Fig. 15-10. Phase portrait of simple har- 
monic motion [Eq. (15-39) with 6 = 0]. 



y = x = — u>oK sin (&>ot + 4>) 

(15-41) 

Since time appears in Eqs. (15-40) and (15-41) merely as a parameter, 
as an alternative form of presenting the answer, x may be plotted as a 
function of x. 

Equations (15-40) and (15-41) are parametric equations of an ellipse. 
Eliminating t yields, as the solution of the original differential equation, 






= 1 



(15-42) 



The starting point of the plot on the x, x plane (hereafter called the phase 
plane) is simply the coordinates given by the initial conditions. Since 
two pieces of information uniquely determine any second-degree curve, 
one and only one ellipse, satisfying Eq. (15-42), may be drawn through 
each point on the plane. The totality of all such curves (Fig. 15-10) is 
called the phase portrait of the system. 

Rotation of the trajectory about the origin is always in a clock- 
wise direction. Counterclockwise rotation would indicate a decreasing 
displacement when the velocity is still positive and an increasing dis- 
placement for negative velocities — an obviously impossible situation. 
Furthermore, these ellipses must cross the x axis at right angles. The x 



494 



OSCILLATIONS 



[Chap. 15 



intercepts are the points at which the velocity changes direction, and 
consequently they must also be the points of the displacement maxima. 
The above arguments lead directly to two additional conclusions: 

1. All closed paths must encircle the origin. 

2. All closed curves correspond to periodic motions. 

The equilibrium point of the system is where x and x vanish. In 
this example only the origin is stable, and it therefore is called a singular 
point. 





(a) (b) 

Fig. 15-11. Phase trajectories of the linear second-degree equation, (a) Damped 
oscillatory response; (b) the aperiodic solution. 

If friction (or damping) is included in the system, then b > and 
Eq. (15-39) has two possible solutions. 

1. The damped oscillatory wave: 

x = Ke~" cos (coi< + <t>) (15-43) 

which exists when the roots are complex conjugates, 

Pi. j = — a ± j"i 

The phase portrait corresponding to Eq. (15-43) and its derivative 
appears in Fig. 15-1 la. As the oscillatory wave damps out, the trajec- 
tory describes a logarithmic spiral about the origin, circling it once 
for each cycle of the cosinusoidal portion of the solution. Since the 
spiral eventually terminates at the origin, this singularity is now called 
a stable focal point. 

2. The damped aperiodic wave which corresponds to the overdamped 

case: 

x = Kitr*' + K&-*' (15-44) 

The damped aperiodic wave of Eq. (15-44) cannot have more than one 



Sec. 15-4] negative-resistance oscillators 495 

point of zero velocity (excluding the trivial point at t = <»). Therefore 
the phase trajectory will intercept the x axis once — it cannot encircle 
the origin. In Fig. 15-116, the origin becomes the stable nodal point 
of the system. 

Isocline Construction. The trajectories that represent the solution of 
the differential equation on the phase plane may be plotted directly from 
the equation by the method of isoclines. These are the locus of all points 
in the phase plane where the curve has the same slope. By making the 
substitution 



Eq. (15-39) may be rewritten 

dy 
dt 



dx 

y = di 



-by - o><?x (15-45) 



But in order to find the slope, Eq. (15-45) will be divided through by y, 
resulting in 

„ dy dy dt , „x . fl . 

S = Tx = diTx=- h - m y < 1W6 > 

Thus all points having the same slope S lie on a straight line through the 
origin, satisfying the equation 

Once the isoclines are drawn, short-line segments of the appropriate 
slope may be marked off along each one, as shown in Fig. 15-12. These 
will serve as guides when drawing the phase portrait. Starting at any 
initial point, a short section of the curve can be drawn having the slope 
defined by the isocline passing through the starting point. From the 
end of the first section, the next section is drawn, and so on, each inter- 
secting the next clockwise isocline with the proper slope until the curve 
is completed. The totality of the individual segments is the actual 
phase trajectory satisfying the initial conditions. 

Such a construction is carried out in Fig. 15-12 for the damped oscilla- 
tory case. For the purpose of illustration, we set co = 1 and b = y±, 
reducing Eq. (15-47) to 

The initial conditions, which give the starting point of the single tra- 
jectory drawn, are x = — 3 and x = 5. 

Returning to Eq. (15-47), we might observe that the condition of an 



496 



OSCILLATIONS 



[Chap. 15 



infinite slope is satisfied along the x axis. This proves the contention 
that the trajectories cross the x axis perpendicularly. 

The most direct path to the stable point would be along an isocline. 
Since the slope of the trajectory is identical with the slope of the isocline, 























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>i 








































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n 




-II 
1 










































o 


■6- 




£/ 
























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% 




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ro 
Ln 


07 




♦7 






















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Hr"" 








s" 




»/ 




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V 


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2T 1 


Initia 

onditio 

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K 


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y£ 


\c 


ns 


















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sJH 






<fr 


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V 




















1 


^ 


^1 


-s?r 




*v. 
















\ 
























' 




^ 


rO> 


fej 








































^ 














































'iK ,? ' 


so- 












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fl ?60 


















N 






























































































































-5 


—t 


\ 


1-3 


-2 


j 


^ 






SO--- 


__2 




3 


^ 




5 


* 


























V 






































































































































































^ 


Jf 






















-3 






























































































~ 










































) 








\ 






































I 










\ 
































/ 






j 










\ 






V 






y 




















/ 






I 










V 




\ 






\ 









Fig. 15-12. Isocline method of constructing a phase trajectory, 
from Eq. (15-47) we see that the necessary condition for such a path is 

S = 



S + b 

But by multiplying out and collecting terms, there results the quadratic 

S 2 + bS + o> 2 = 

Except that it is a function of S, this quadratic is identical with the 
characteristic equation from which the roots of the original differential 
equation (15-39) are found. 
Straight lines having a slope equal to the roots are feasible only when 



Sec. 15-4] negative-besistance oscillators 497 

these roots are real, i.e., when the response is aperiodic. Two such 
lines exist on each phase plane as shown in Fig. 15-116. As they head 
toward the origin, all trajectories approach the line corresponding to the 
smallest root. It predominates, since it represents the long time decay 
of the system. The larger root determines the initial rise, and its isocline 
separates the trajectories that finally approach the origin in the second 
quadrant from those that terminate in the fourth quadrant. 

Unstable Equilibrium. If energy is supplied to the system at the proper 
rate, for example, by connecting the negative driving-point resistance 





(a) 
Fig. 15-13. Trajectories for negative damping. 
(b) increasing aperiodic solution. 



(b) 
(a) Increasing oscillatory response; 



across a GLC parallel circuit, the amplitude would increase rather than 
decrease with time. In the region where the circuit may be approxi- 
mated by a quasi-linear equation, i.e., before limiting, the trajectories 
would diverge from the singularity at the origin, rotating clockwise as 
they do so. The two cases of most interest are shown in Fig. 15-13. 
The increasing spiral represents the build-up of an almost sinusoidal 
oscillator, and the aperiodic trajectory illustrates the switching path of a 
multivibrator. If the system has some initial energy, the single tra- 
jectory of interest would be the one passing through the point on the 
phase plane corresponding to the initial conditions. In the case of an 
increasing exponential, the trajectories will approach the isocline cor- 
responding to the larger root of the second-order equation, regardless 
of how they diverge from the origin. If we are interested in the response 
very far away from the point corresponding to the initial conditions, 
then this particular isocline would closely approximate the path. Of 
course, nonlinearities in the system may distort the trajectories long 
before they ever reach this asymptote. 

Period and Waveshape. Topological methods may also be employed 
to convert the trajectory on the phase plane into the equivalent time- 
varying signal. We can construct a small line segment in the time domain 



498 



OSCILLATIONS 



[Chap. 15 



having the slope and position given by the corresponding point on the 
phase plane. To the end of the first one, a second segment is added, 
then a third, and so on. As we work our way along the trajectory, the 
totality of all segments will give a rough approximation to the waveshapes 
generated. This process is, of course, quite tedious, but necessary where 
simpler methods cannot yield a satisfactory solution. 



t ' 

X 


-1 
-2 



























































































































































































































































































































































































t, sec 


j 








) 




! 


t 




\ 




i 






l 


h 


3 


c 


> ' 


/ 






































/ 






































/ 






































/ 






































1 































































































































































































-5 
Fia. 15-14. Waveshape constructed from the trajectory of Fig. 15-12. 

Figure 15-14 shows just such a construction for slightly more than one 
complete encirclement of the origin by the trajectory of Fig. 15-12. The 
nitial point of the trajectory was chosen as the start of the construction. 
Wherever the slope is large, large increments of position are permissible 
over the associated line segment. Since this section of the curve does not 
contribute much toward the period, relatively gross approximations 
are justified. But when the operating point is moving very slowly, the 
positional increments must be quite close together. To further reduce 
the timing error, the average slope over each segment of the trajectory 
was used in the construction of Fig. 15-14. 

The timing accuracy of this construction is very poor, and therefore it 
should be used mainly to identify the waveshape. In the example shown, 
the time for one complete cycle is almost 1 sec too long. 



Sec. 15-5] negative-resistance oscillators 499 

The time required for the operating point to pass between any two 
points on the trajectory is found from the line integral: 

<f-dx = <f^dx = (f'dl=T b -T a (15-48) 

In general, Eq. (15-48) is evaluated by means of a graphical or a 
numerical integration. For the complete period of any closed contour on 
the phase plane, the integral becomes 

T = <£-dx (15-49) 

and it is taken over the complete trajectory. 

15-6. Lienard Diagram. As the first step in finding the topological 
solution of the Van der Pol nonlinear differential equation, we shall 
normalize the basic equation (15-8) by dividing through by wo 2 ; this 
leads to the result 

Since the variables of interest are the voltage across the terminals of the 
VNLR and its derivative, there should not be any ambiguity or confusion 
of terms if we redefine v = dv/d(o>ot) and rewrite Eq. (15-50) : 

v + e'f(v)i> + v = (15-51) 

And following the procedure outlined in Sec. 15-4, the slope of the 
trajectory, at each point in the phase plane, will be given by 

« - £ - - !m i ±I WD 

The isoclines defined by Eq. (15-52) are no longer straight lines, but 
are curves which depend on the nature of the nonlinearity. With 
any given volt-ampere characteristic, f(v) may be evaluated for each 
value of v. The isoclines are finally constructed, if necessary by plotting 
individual points. 

As this process is extremely tedious, we shall instead turn to the simpler 
construction developed by A. Lienard. By means of a linear transforma- 
tion, from the phase plane to the Li6nard plane, the problem becomes one 
of finding the normal to the trajectory and then simply striking an arc 
to find a segment of the topographical portrait. The transformation 
chosen is 

z = i> + e'F(v) (15-53) 

where /(*) = ^ (15-54) 



500 oscillations [Chap. 15 

By differentiating Eq. (15-53) with respect to v and rearranging terms, 
an alternative expression for the isoclines results: 

„ di) dz ... , 
S = dv = dv ~ * m 

Substituting S into Eq. (15-52) leads to 

dz _ _ v 
dv v 

From the linear transformation of Eq. (15-53), the slope of the trajectory 




Fig. 15-15. Illustrating the means of evaluating F (v) .from the volt-ampere character- 
istics. 



on the Lienard plane finally becomes 

dz v 

dv~ z - e'F(v) 

and the slope of the normal to the trajectory is given by 

z - e'F(v) 



N = 



dv 
dz 



(15-55) 



(15-56) 



Before we can construct the trajectories we must evaluate t'F(v). 
For the basic circuit of Fig. 15-1, 



dF(v) 



= f(v) =G + 



di 



(15-57) 



dv Jy "' " ' dv 

Integrating both sides with respect to v yields 

F(w) = Gv + i = i - {-Gv) 

where v and i are measured from the Q point on the volt-ampere char- 
acteristics and where — Gv is the conductive load line passing through the 
Q point. Equation (15-57) says that F(v) is the difference in current 
between the load line and the characteristic (as shown by the vectors in 
Fig. 15-15). 



Sec. 15-5] 



NEGATIVE-RESISTANCE OSCILLATORS 



501 



In replotting F(v), the ordinate would also be multiplied by «' to scale 
the curve. This factor is dependent on the design of the tank and may, 
in fact, be considered a defining parameter. 



cooC 



(15-58) 



The degree of the circuit nonlinearity is proportional to «'; the system 
becomes quasi-linear for very small values of «' and grossly nonlinear for 
very large values. We conclude that for an almost sinusoidal oscillator, 



. 


z-i+e'F(v) 


Pi 


Pa 


P 3 


A'FXv) 


/ ^v r " 










\ 


/ r3 






/I3 V 


r 2 






\ 


r l 




*1 



Fig. 16-16. Construction of the Lienard diagram. 

L/C should be made as small as possible consistent with the other design 
requirements. This supports the arguments made earlier with respect to 
Figs. 15-2 and 15-3. 

The final plot of t'F(v), which serves as the basis for the construction 
of the Lienard diagram, appears in Fig. 15-16. The procedure followed 
in constructing first the normal and then the trajectory is based on 
Eq. (15-56). The various steps illustrated in Fig. 15-16 are: 

1. Plot the nonlinearity e'F(v), following the construction outlined 
above, on the Lienard plane. This plot cannot be scaled in either 
coordinate but must be drawn 1:1. 

2. At any point on the plane pi, drop a perpendicular to e'F(v) at q\. 

3. Draw a line from gi to the z axis at ri. 

4. The vertical line segment 

Piqi = 21 - t'F{vi) 
at that particular value of z x and V\. The horizontal line segment 

qiri = vi 



502 



OSCILLATIONS 



[Chap. 15 



Hence, from Eq. (15-56), the hypotenuse of the triangle ripiqi is the 
normal to the trajectory passing through point p\. 

5. Strike small arcs intersecting line segment piqi using point r t as 
the center. These arcs are segments of the various trajectories on the 
Lienard plane. 

6. Starting at any initial point on the plane, strike an arc; from the 
end of this arc strike another one; and then continue the construction 
from one point to the next in a clockwise direction until the trajectories 
converge into a closed contour. 

A construction for four points appears in Fig. 15-16, and complete 
trajectories, cycling to the closed curve representing a periodic solution, 




Fig. 15-17. Lienard-plane construction of the trajectory and the limit cycle and the 
phase-plane limit cycle. 



Sec. 15-5] 



NEGATTVE-BESISTANCE OSCILLATORS 



503 



appear in Figs. 15-17 and 15-18. In Fig. 15-17 three different starting 
points, two close to the origin and one at z = 30, v = 0, were chosen. 
By following the procedure outlined above, three trajectories are con- 
structed, and these all eventually form the same closed contour encircling 
the origin. This curve represents the only periodic solution of the sys- 
tem, and it is called a limit cycle. The path leading to the limit cycle, 















, 


2 






















































s 


.'" 






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"N 


















, 


V 


























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1 
















\ 














1 








I 

"5B~ 


nstal 


>le lin 

I 


lit cy 


cle 


* 


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— 1- 
\ 
\ 
















\ 












\ 


V 






30 










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,,' 


*~" 


"**< 


\ 








T l 


t 




















V 






l 


lr 


















Js 


n ] 


r / 
0/2 


0/3 


4 




6 


70 . 














sV. 


S 


t-'y 










V 






















f-St 


able 


imit ( 


ycle 



































Fio. 16-18. Lienard-plane construction illustrating the conditions yielding both stable 
and unstable limit cycles. 



from any starting point, is the transient portion of the solution. Two 
trajectories in Fig. 15-17 show the exponential build-up. One, which 
starts with excessive amplitude, damps down to the final steady-state 
solution. 

The plot on the Lienard plane can be converted into a phase portrait 
by performing the inverse transformation 

v = z- e'F(v) 

which involves subtracting from each point on the Lienard trajectory the 
value of t'F(v). This process is also carried out in Fig. 15-17. Because 
of the large degree of nonlinearity, it leads to the grossly distorted contour 
shown. 
The phase portrait, as well as the Lienard plot of an oscillatory system, 



504 oscillations [Chap. 15 

will exhibit one or more limit cycles. If all adjacent trajectories converge 
on one contour (as in Fig. 15-17), then it represents a stable oscillation. 

Every unstable equilibrium condition is associated with some closed 
contour from which the trajectories representing the transient response 
diverge. It follows that stable and unstable limit cycles separate each 
other. The origin may also be regarded as a solution which may be 
either stable or unstable, depending on the nature of the trajectories 
terminating on it. If the origin is unstable, the circuit is self-starting. 
If the origin is stable, the circuit must be triggered past the adjacent 
unstable limit cycle. 

Figure 15-18 illustrates the nature of the unstable limit cycle, which 
arises when the negative-resistance device is normally biased in its dis- 
sipative region near the knee of the volt-ampere characteristics. For 
small signals, the circuit is stable and all trajectories cycle toward the 
origin. However, after the signal becomes large enough so that the 
circuit exhibits an average negative resistance, the build-up will proceed 
toward the next stable limit cycle. These limit cycles correspond to the 
fundamental amplitudes Et and E f in the describing-function solutions 
of Figs. 15-6 and 15-7. 

The shape of the limit cycle, which depends on the size of «' for 
any given volt-ampere characteristic and frequency, indicates the 
nature of the time response. When e' is small, the limit cycle approaches 
a circle or ellipse and the signal generated is almost sinusoidal (Fig. 
15-19o). When e' is large, the closed trajectory becomes almost rec- 
tangular and the time response is that of a relaxation oscillator (Fig. 
15-196). 

Since the current through a capacitor is proportional to the derivative 
of the applied voltage, the phase portrait is readily obtained experi- 
mentally. The VNLR voltage is applied to the x plates of a cathode-ray 
oscilloscope and a voltage proportional to the capacitor current to the 
y deflection circuit. 

At this point we might observe that the various topological construc- 
tions discussed in this chapter may also be applied to the solution of the 
negative-resistance switching circuits of Chap. 11. At the low fre- 
quencies of operation, the external inductance in the VNLR switching 
circuit predominates. This corresponds to a very large value of t'. 
A trajectory similar to that shown in Fig. 15-196 would be obtained. 
Analogous response appears in the CNLR switching circuit. At the 
higher frequencies, the effective internal energy-storage element of the 
device plays a more pronounced role, reducing d and introducing curva- 
ture into the trajectory. Thus the phase portrait explains the changes 
in the behavior of the multivibrator with frequency and even its almost 
sinusoidal time response when operating at the elevated frequencies. 



Sec. 15-6] negative-resistance oscillators 505 

Furthermore, the phase- or Lienard-plane construction can be used to 
illustrate the transient switching path between the two states of a bistable 
circuit. Since the stable points are the nodal points of the system, two 
constructions would be necessary, one with each singular point as the 
origin. 





Wt 



(a) (b) 

Fig. 15-19. Limit cycles and time response for two values of e'. (a) «' very small — 
operation as an almost sinusoidal oscillator; (6) t large — operation as a relaxation 
oscillator. 



16-6. Summary. The complete solution of the nonlinear oscillator, 
i.e., the waveshape, amplitude, and period, cannot be arrived at by 
either the topological or the analytic method alone. These two tech- 
niques should not be regarded as independent, but rather as comple- 
mentary: where one is weak, the other is strong. By combining the 
results, we are able to obtain almost any required information as to the 
behavior of the system, information which cannot readily be obtained from 
either method individually. 

To find the frequency of oscillation, including the depression due to the 
harmonic content of the output, we must turn to the analytic solution 
where an explicit expression was derived [Eq. (15-30)]. However, to 
evaluate this equation, we must have some knowledge as to the wave- 
shape produced. If this is to be obtained analytically, then the non- 
linearity must be expressed as a power series. After the fundamental 
amplitude is found from the describing function, the harmonic content is 
evaluated by substituting back into this series. In simple cases, for 



506 oscillations [Chap. 15 

example, where the volt-ampere characteristic is almost symmetrical, the 
accuracy requirements are satisfied by approximating the nonlinearity 
by a cubic polynomial. The coefficients are relatively easy to evaluate. 
However, in most cases a fifth- or higher-degree equation is necessary 
for an adequate description, and it becomes extremely tedious to solve 
for the coefficients. Finally, we should note that the analytic method 
does not give the waveshape of the output directly, but only as a Fourier 
series. 

On the other hand, the topological solution yields the peak-to-peak 
amplitude of the stable oscillation quite rapidly but supplies almost no 
accurate information as to the frequency of operation. A good approxi- 
mation to the waveshape generated is easily constructed. From it one 
could find the relative harmonic content of the output either by a graph- 
ical integration or by some schedule method. 

Suppose that we assume that the period of the waveshape constructed 
can be scaled to satisfy the frequency found from the analytic expression. 
Then, from the topological construction, we obtain the waveshape and 
the harmonic content and use this to find the corrected frequency. Thus 
each mode of solution is employed where it best serves and, without 
actually arriving at an analytic expression for the output voltage, we are 
able to solve the problems posed. 



PROBLEMS 

15-1. (a) Prove that the vacuum-tube Colpitts oscillator may also be treated as a 
negative-resistance circuit. 

(b) Convert the negative conductance appearing from plate to ground (found in 
part a) to an equivalent negative conductance across the complete tuned circuit. 

(c) State the bounding values of g m for which this circuit will produce an almost 
sinusoidal output. 

15-2. A transistor Hartley oscillator is to be converted into its negative-impedance 
equivalent circuit. Assume that the transistor has a very large base-to-collector 
transconductance and that the two coils are not coupled. Make all reasonable 
approximations. Find the limits of r c /r' n for which the circuit will function as an 
oscillator. 

15-3. A unijunction transistor (Chap. 11) having 7 = 3, ni = 300, and m = 
200 is biased in its active region and is used as the active element in the CNLR 
oscillator of Fig. 15-5. The circuit is designed to operate at 50 kc with a tuned cir- 
cuit that has a loaded Q = 10. Specify the limiting values of L and C for which this 
circuit will function as an oscillator. 

15-4. (a) Specify the slopes and bounds of the regions of Fig. 15-4d if the tube has 
n = 50, r p = 20 K, and r c = and if the conductance of the tank is 10~ 4 mho. The 
plate supply is 100 volts, and the tube is biased in the center of its active region. 
State all approximations made in the course of the analysis. 

(b) Will the positive grid loading have any effect other than the reduction in gain? 
Explain. 



NEGATIVE-RESISTANCE OSCILLATORS 



507 



(c) What limitation must be imposed on the other tank parameters for proper cir- 
cuit operation if M = 0.2L? 

16-6. Two negative-resistance characteristics have power-series expansions about 
the Q point given by 

1. i. = 3.0» - 0.2v» + O.OOIv* ma. 

2. u = -3.0v + 0.2»» + O.OOlw 6 ma. 

(a) Plot the current describing functions of these two curves on the same graph 
and find the amplitude of oscillations if G = 2 X lO -4 mho. ; if G = 1 X lO" 8 mho. 

(6) Repeat part a with respect to the conductance describing function. 

16-6. This problem is designed to investigate the response of various negative- 
resistance oscillators from a consideration of the power-series expansion. Assume 
that the terms of interest are 

»« = <*i» + a 2 f * + a 3 v' + a,v* + a,i» s 

(a) Prove that in order for an oscillator to be self-starting, 

a, < -G 
(6) Prove that 

a b >0 

for any values of the other coefficients. 

(c) Under what conditions will the describing function have a minimum at some 
nonzero value of Ei? State the answer in terms of the relative size of the coefficients. 

(<f) Under what conditions will the describing function have both a maximum and 
minimum value of i for some nonzero value of Ei"! 

16-7. Figure 15-20 shows the piecewise-linear volt-ampere characteristic of a semi- 
conductor device. Find the approximating cubic polynomial holding when the quies- 
cent point is in the center of the negative-resistance region. Solve for the amplitude 
of osculation when R = 400 and y/L/C - 1,000. 




Fig. 15-20 

16-8. Express the frequency deviation in terms of the harmonic current components 
of the VNLR. Give the answer in a form analogous to Eq. (15-30). State the rea- 
son for all approximations made in the course of the solution. 

16-9. Consider a negative-resistance oscillator using a device described by the 
polynomial 

i = — 2.0w + 0.5»' ma 

(a) Find the amplitude of the fundamental term when G = 1.0 millimho. 

(b) Calculate the amplitude of the third-harmonic current component with the 
assumed sinusoidal excitation found in part a. 



508 oscillations [Chap. 15 

(c) When C = 100 it/A and when wo = 10 8 , evaluate the depression in frequency 
due to the harmonic voltage developed. 

(d) Repeat the above calculations for O — 1.8 millimhos. 

16-10. Use Eq. (15-28) to calculate the approximate depression in frequency if 
the distortion is great enough so that the terminal voltage is almost a square wave. 
Even though the answer will be far from exact, it will still give a rough order of 
magnitude. 

15-11. An oscillator with a nominal resonant frequency of at, = 10" and a signal 
amplitude of 20 volts has a 5 per cent third-harmonic and a 1 per cent fifth-harmonic 
component. The tuned circuit has a Q of 50 at an,- 

(a) By what percentage will the frequency be depressed? 

(b) If C = 100 ii/xi, what is the phase angle of the fundamental? 

(c) What is the amplitude of the injected seventh-harmonic component that will 
just return the frequency to o> ? 

(d) The frequency is to be increased to 1.03oj by injecting a second-harmonic 
signal. What is the necessary amplitude of the synchronizing signal? Calculate 
the new phase angle at the fundamental. 

15-12. Find the phase trajectory of the following single-order system under the 
conditions given: 

x + bx = 

Plot all trajectories on the same graph and justify any jumps in x that appear. 

(a) b = -5 and 2(0) = 1. 

(b) b H and x(0~) V 2 , 2(0-) = 3. 

(c) 6 = +2 and x(0) = 10. 

(<*) b = +H and 2(0") = 10, 2(0") 10. 

Find the time response for parts b and d by means of a graphical construction from 
the phase plane. 

15-13. Plot the phase portrait of the following equation on a normalized phase 
plane; i.e., the coordinates are i/w and x. 

x+ 2Aw<>x + «o ! 2 = 

(a) Set A = 1.1. 

(6) Set A = 3. 

Pay particular attention to at least one path that falls between the straight-line 
trajectories. 

15-14. Repeat Prob. 15-13 for 

(a) A = -1.1. 

(6) A = -3. 

16-15. The behavior of an undamped pendulum in the vicinity of its unstable 
point is defined by 

x — bx = 

where b > 0. In this case the origin is called a saddle point. 

(a) Prove that the trajectories are hyperbolas. 

(b) Plot two pairs of trajectories when b = 4. Sketch the asymptotes and indicate 
the direction of rotation of all curves. 

16-16. Assume that the plot drawn in Fig. 15-17 corresponds to t' = 1. We now 
wish to consider the response when t' =0.1 and when e' = 10. 

(a) Construct the limit cycle on the Lienard plane for these two cases. 

(b) Sketch the approximate waveshapes produced. 

(c) By how much does the peak-to-peak amplitude change as e' increases from 0. 1 
to 1 to 10? 



NEGATIVE-RESISTANCE OSCILLATORS 



509 



15-17. Accurately construct, out of small line segments, the time-varying signal 
represented by the limit cycle of Fig. 15-17. 

15-18. Plot a Lienard-plane trajectory and the equivalent phase-plane portrait for 
the oscillator employing a device having the piecewise-linear volt-ampere character- 
istic of Fig. 15-21. The quiescent point is located at V„ — 50 volts and I„ = 5 ma, 
and the tank parameters are VX/C = 3,000 and R = 5,000 ohms. Sketch the 
waveshape. 




40 60 

Fig. 15-21 



120 volts 



15-19. The load resistance of Prob. 15-18 is changed to 10 K and the Q point to 
Vo = 80 volts and I„ = 3 ma. 

(o) Plot at least two Lienard-plane trajectories showing the transient decay toward 
the stable point. 

(b) Is it possible to make this circuit oscillate? Give the expected peak-to-peak 
output if it is possible. If it is impossible, justify your answer. 

15-20. Solve Prob. 15-7 by the method of the Lienard plane. 



BIBLIOGRAPHY 

Andronow, A. A., and C. E. Chaikin: "Theory of Oscillations," Princeton University 

Press, Princeton, N.J., 1949. 
Belevitch, V.: "Theorie des circuits non-lineaires en regime alternatif," Gauthier- 

Villars, Paris, 1959. 
Edson, W. A.: "Vacuum-tube Oscillators," John Wiley & Sons, Inc., New York, 1953. 
Kryloff, N., and N. Bogoliuboff: "Introduction to Non-linear Mechanics," Princeton 

University Press, Princeton, N.J., 1943. 
le Corbeiller, P. : The Nonlinear Theory of the Maintenance of Oscillations, J. IEE 

{London), vol. 79, pp. 361-378, 1936. 
Lienard, A.: Etude des oscillations entretenus, Rev. gin. ilec, vol. 23, pp. 901-946, 

1928. 
Minorsky, N.: "Introduction to Nonlinear Mechanics," J. W. Edwards, Publisher, 

Inc., Ann Arbor, Mich., 1947. 
Oser, E. A., R. O. Enders, and R. P. Moore, Jr.: Transistor Oscillators, RCA Rev., 

vol. 13, pp. 369-385, September, 1952. 
van der Pol, B. : The Nonlinear Theory of Electric Oscillations, Proc. IRE, vol. 22, 

no. 9, pp. 1051-1086, 1934. 



NAME INDEX 



Aaron, M. R., 384 
Abraham, H., 288 
Anderson, A. E., 356 
Anderson, F. B., 476 
Andronow, A. A., 509 
Angelo, E. J., Jr., 66, 143 



Bates, L. F., 427 
Beale, I. E. A., 356 
Belevitch, V., 509 
Benjamin, R., 384 
Bloch, E., 288 
Bode, H. W., 476 
Bogoliuboff, N., 509 
Bollman, J. H., 476 
Boothroyd, A. R., 221 
Bothwell, F. E., 476 
Brenner, E., 25 
Briggs, B. H., 205 
Brown, D. R., 95 



Edson, W. A., 476, 509 
Enders, R. O., 509 
Esaki, L., 338, 356 



Farley, B. G., 356 
Feinberg, R., 288 



Gibbons, J. F., 356 
Ginzton, E. L., 476 
Glegg, K., 314 
Goodrich, H. C, 243 
Guillemin, E. A., 25 



Hall, R. N., 356 
Herzo